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Theorem ac6s6 33609
 Description: Generalization of the Axiom of Choice to classes, moving the existence condition in the consequent. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
Hypotheses
Ref Expression
ac6s6.1 𝑦𝜓
ac6s6.2 𝐴 ∈ V
ac6s6.3 (𝑦 = (𝑓𝑥) → (𝜑𝜓))
Assertion
Ref Expression
ac6s6 𝑓𝑥𝐴 (∃𝑦𝜑𝜓)
Distinct variable groups:   𝜑,𝑓   𝑥,𝑦   𝑥,𝐴,𝑓   𝑦,𝑓   𝐴,𝑓
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦,𝑓)   𝐴(𝑦)

Proof of Theorem ac6s6
StepHypRef Expression
1 hbe1 2018 . . . . . 6 (∃𝑦𝜑 → ∀𝑦𝑦𝜑)
2 iftrue 4064 . . . . . . 7 (∃𝑦𝜑 → if(∃𝑦𝜑, {𝑦𝜑}, V) = {𝑦𝜑})
32abeq2d 2731 . . . . . 6 (∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑))
41, 3exbidh 1791 . . . . 5 (∃𝑦𝜑 → (∃𝑦 𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ∃𝑦𝜑))
54ibir 257 . . . 4 (∃𝑦𝜑 → ∃𝑦 𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V))
6 vex 3189 . . . . . 6 𝑦 ∈ V
76exiftru 1888 . . . . 5 𝑦 𝑦 ∈ V
81hbn 2142 . . . . . 6 (¬ ∃𝑦𝜑 → ∀𝑦 ¬ ∃𝑦𝜑)
9 iffalse 4067 . . . . . . 7 (¬ ∃𝑦𝜑 → if(∃𝑦𝜑, {𝑦𝜑}, V) = V)
109eleq2d 2684 . . . . . 6 (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V))
118, 10exbidh 1791 . . . . 5 (¬ ∃𝑦𝜑 → (∃𝑦 𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ∃𝑦 𝑦 ∈ V))
127, 11mpbiri 248 . . . 4 (¬ ∃𝑦𝜑 → ∃𝑦 𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V))
135, 12pm2.61i 176 . . 3 𝑦 𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V)
1413rgenw 2919 . 2 𝑥𝐴𝑦 𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V)
15 nfe1 2024 . . . 4 𝑦𝑦𝜑
16 ac6s6.1 . . . 4 𝑦𝜓
1715, 16nfim 1822 . . 3 𝑦(∃𝑦𝜑𝜓)
18 ac6s6.2 . . 3 𝐴 ∈ V
19 ac6s6.3 . . . . . 6 (𝑦 = (𝑓𝑥) → (𝜑𝜓))
20 id 22 . . . . . . . . . . . . . . 15 𝜑 → ¬ 𝜑)
2120a1i 11 . . . . . . . . . . . . . 14 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ 𝜑 → ¬ 𝜑))
22 ax-1 6 . . . . . . . . . . . . . . . . . . 19 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ 𝜑 → ¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))))))
23 tsim3 33568 . . . . . . . . . . . . . . . . . . . 20 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) ∨ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))))))
2423a1d 25 . . . . . . . . . . . . . . . . . . 19 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ 𝜑 → (¬ ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) ∨ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))))))
2522, 24cnf2dd 33522 . . . . . . . . . . . . . . . . . 18 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ 𝜑 → ¬ ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))))
26 tsim3 33568 . . . . . . . . . . . . . . . . . . 19 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))) ∨ ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))))
2726a1d 25 . . . . . . . . . . . . . . . . . 18 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ 𝜑 → (¬ (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))) ∨ ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))))))
2825, 27cnf2dd 33522 . . . . . . . . . . . . . . . . 17 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ 𝜑 → ¬ (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))))
29 tsim2 33567 . . . . . . . . . . . . . . . . . 18 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (∃𝑦𝜑 ∨ (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))))
3029a1d 25 . . . . . . . . . . . . . . . . 17 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ 𝜑 → (∃𝑦𝜑 ∨ (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))))
3128, 30cnf2dd 33522 . . . . . . . . . . . . . . . 16 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ 𝜑 → ∃𝑦𝜑))
32 tsim2 33567 . . . . . . . . . . . . . . . . . 18 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) ∨ ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))))
3332a1d 25 . . . . . . . . . . . . . . . . 17 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ 𝜑 → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) ∨ ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))))))
3425, 33cnf2dd 33522 . . . . . . . . . . . . . . . 16 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ 𝜑 → (∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑))))
3531, 34mpdd 43 . . . . . . . . . . . . . . 15 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ 𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)))
36 tsbi4 33572 . . . . . . . . . . . . . . . 16 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → ((¬ 𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ∨ 𝜑) ∨ ¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)))
3736a1d 25 . . . . . . . . . . . . . . 15 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ 𝜑 → ((¬ 𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ∨ 𝜑) ∨ ¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑))))
3835, 37cnfn2dd 33524 . . . . . . . . . . . . . 14 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ 𝜑 → (¬ 𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ∨ 𝜑)))
3921, 38cnf2dd 33522 . . . . . . . . . . . . 13 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ 𝜑 → ¬ 𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V)))
40 tsim3 33568 . . . . . . . . . . . . . . . . 17 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))) ∨ (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))))
4140a1d 25 . . . . . . . . . . . . . . . 16 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ 𝜑 → (¬ (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))) ∨ (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))))
4228, 41cnf2dd 33522 . . . . . . . . . . . . . . 15 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ 𝜑 → ¬ (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))
43 tsim3 33568 . . . . . . . . . . . . . . . 16 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)) ∨ (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))
4443a1d 25 . . . . . . . . . . . . . . 15 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ 𝜑 → (¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)) ∨ (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))))
4542, 44cnf2dd 33522 . . . . . . . . . . . . . 14 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ 𝜑 → ¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))
46 tsbi2 33570 . . . . . . . . . . . . . . 15 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → ((𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ∨ (∃𝑦𝜑𝜓)) ∨ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))
4746a1d 25 . . . . . . . . . . . . . 14 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ 𝜑 → ((𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ∨ (∃𝑦𝜑𝜓)) ∨ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))
4845, 47cnf2dd 33522 . . . . . . . . . . . . 13 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ 𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ∨ (∃𝑦𝜑𝜓))))
4939, 48cnf1dd 33521 . . . . . . . . . . . 12 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ 𝜑 → (∃𝑦𝜑𝜓)))
50 tsim2 33567 . . . . . . . . . . . . . . . . . . 19 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (𝑦 = (𝑓𝑥) ∨ (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))
5150a1d 25 . . . . . . . . . . . . . . . . . 18 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ 𝜑 → (𝑦 = (𝑓𝑥) ∨ (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))))
5242, 51cnf2dd 33522 . . . . . . . . . . . . . . . . 17 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ 𝜑𝑦 = (𝑓𝑥)))
53 simplim 163 . . . . . . . . . . . . . . . . 17 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (𝑦 = (𝑓𝑥) → (𝜑𝜓)))
5452, 53syld 47 . . . . . . . . . . . . . . . 16 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ 𝜑 → (𝜑𝜓)))
55 tsbi3 33571 . . . . . . . . . . . . . . . . 17 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → ((𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑𝜓)))
5655a1d 25 . . . . . . . . . . . . . . . 16 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ 𝜑 → ((𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑𝜓))))
5754, 56cnfn2dd 33524 . . . . . . . . . . . . . . 15 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ 𝜑 → (𝜑 ∨ ¬ 𝜓)))
5821, 57cnf1dd 33521 . . . . . . . . . . . . . 14 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ 𝜑 → ¬ 𝜓))
59 tsim1 33566 . . . . . . . . . . . . . . . 16 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → ((¬ ∃𝑦𝜑𝜓) ∨ ¬ (∃𝑦𝜑𝜓)))
6059a1d 25 . . . . . . . . . . . . . . 15 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ 𝜑 → ((¬ ∃𝑦𝜑𝜓) ∨ ¬ (∃𝑦𝜑𝜓))))
6160or32dd 33525 . . . . . . . . . . . . . 14 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ 𝜑 → ((¬ ∃𝑦𝜑 ∨ ¬ (∃𝑦𝜑𝜓)) ∨ 𝜓)))
6258, 61cnf2dd 33522 . . . . . . . . . . . . 13 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ 𝜑 → (¬ ∃𝑦𝜑 ∨ ¬ (∃𝑦𝜑𝜓))))
6331, 62cnfn1dd 33523 . . . . . . . . . . . 12 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ 𝜑 → ¬ (∃𝑦𝜑𝜓)))
6449, 63contrd 33528 . . . . . . . . . . 11 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → 𝜑)
6564a1d 25 . . . . . . . . . 10 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ ⊥ → 𝜑))
66 ax-1 6 . . . . . . . . . . . . . . 15 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ ⊥ → ¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))))))
6723a1d 25 . . . . . . . . . . . . . . 15 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ ⊥ → (¬ ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) ∨ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))))))
6866, 67cnf2dd 33522 . . . . . . . . . . . . . 14 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ ⊥ → ¬ ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))))
6926a1d 25 . . . . . . . . . . . . . 14 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ ⊥ → (¬ (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))) ∨ ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))))))
7068, 69cnf2dd 33522 . . . . . . . . . . . . 13 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ ⊥ → ¬ (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))))
7129a1d 25 . . . . . . . . . . . . 13 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ ⊥ → (∃𝑦𝜑 ∨ (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))))
7270, 71cnf2dd 33522 . . . . . . . . . . . 12 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ ⊥ → ∃𝑦𝜑))
7332a1d 25 . . . . . . . . . . . . 13 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ ⊥ → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) ∨ ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))))))
7468, 73cnf2dd 33522 . . . . . . . . . . . 12 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ ⊥ → (∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑))))
7572, 74mpdd 43 . . . . . . . . . . 11 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ ⊥ → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)))
76 tsbi3 33571 . . . . . . . . . . . 12 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → ((𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ∨ ¬ 𝜑) ∨ ¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)))
7776a1d 25 . . . . . . . . . . 11 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ ⊥ → ((𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ∨ ¬ 𝜑) ∨ ¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑))))
7875, 77cnfn2dd 33524 . . . . . . . . . 10 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ ⊥ → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ∨ ¬ 𝜑)))
7965, 78cnfn2dd 33524 . . . . . . . . 9 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ ⊥ → 𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V)))
8040a1d 25 . . . . . . . . . . . . . . . 16 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ ⊥ → (¬ (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))) ∨ (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))))
8170, 80cnf2dd 33522 . . . . . . . . . . . . . . 15 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ ⊥ → ¬ (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))
8250a1d 25 . . . . . . . . . . . . . . 15 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ ⊥ → (𝑦 = (𝑓𝑥) ∨ (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))))
8381, 82cnf2dd 33522 . . . . . . . . . . . . . 14 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ ⊥ → 𝑦 = (𝑓𝑥)))
8483, 53syld 47 . . . . . . . . . . . . 13 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ ⊥ → (𝜑𝜓)))
85 tsbi4 33572 . . . . . . . . . . . . . 14 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → ((¬ 𝜑𝜓) ∨ ¬ (𝜑𝜓)))
8685a1d 25 . . . . . . . . . . . . 13 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ ⊥ → ((¬ 𝜑𝜓) ∨ ¬ (𝜑𝜓))))
8784, 86cnfn2dd 33524 . . . . . . . . . . . 12 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ ⊥ → (¬ 𝜑𝜓)))
8865, 87cnfn1dd 33523 . . . . . . . . . . 11 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ ⊥ → 𝜓))
8988a1dd 50 . . . . . . . . . 10 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ ⊥ → (∃𝑦𝜑𝜓)))
90 tsbi1 33569 . . . . . . . . . . . 12 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → ((¬ 𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ∨ ¬ (∃𝑦𝜑𝜓)) ∨ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))
9190a1d 25 . . . . . . . . . . 11 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ ⊥ → ((¬ 𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ∨ ¬ (∃𝑦𝜑𝜓)) ∨ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))
9291or32dd 33525 . . . . . . . . . 10 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ ⊥ → ((¬ 𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ∨ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))) ∨ ¬ (∃𝑦𝜑𝜓))))
9389, 92cnfn2dd 33524 . . . . . . . . 9 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ ⊥ → (¬ 𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ∨ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))
9479, 93cnfn1dd 33523 . . . . . . . 8 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ ⊥ → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))
9543a1d 25 . . . . . . . . 9 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ ⊥ → (¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)) ∨ (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))))
9681, 95cnf2dd 33522 . . . . . . . 8 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → (¬ ⊥ → ¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))
9794, 96contrd 33528 . . . . . . 7 (¬ ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))) → ⊥)
9897efald2 33506 . . . . . 6 ((𝑦 = (𝑓𝑥) → (𝜑𝜓)) → ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))))
9919, 98ax-mp 5 . . . . 5 ((∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝜑)) → (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))
1003, 99ax-mp 5 . . . 4 (∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))
1016a1i 11 . . . . . . 7 (¬ ∃𝑦𝜑𝑦 ∈ V)
102 id 22 . . . . . . . . . . . 12 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → ¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))))
103 tsim2 33567 . . . . . . . . . . . . . . 15 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → (¬ ∃𝑦𝜑 ∨ (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤))))
104103ord 392 . . . . . . . . . . . . . 14 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → (¬ ¬ ∃𝑦𝜑 → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤))))
105104a1dd 50 . . . . . . . . . . . . 13 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → (¬ ¬ ∃𝑦𝜑 → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))))
106105a1dd 50 . . . . . . . . . . . 12 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → (¬ ¬ ∃𝑦𝜑 → ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤))))))
107102, 106mt3d 140 . . . . . . . . . . 11 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → ¬ ∃𝑦𝜑)
108107a1d 25 . . . . . . . . . 10 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → (¬ ⊥ → ¬ ∃𝑦𝜑))
109 simplim 163 . . . . . . . . . 10 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → (¬ ∃𝑦𝜑𝑦 ∈ V))
110108, 109syld 47 . . . . . . . . 9 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → (¬ ⊥ → 𝑦 ∈ V))
111 tsim2 33567 . . . . . . . . . . . . . 14 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) ∨ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))))
112111ord 392 . . . . . . . . . . . . 13 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → (¬ (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))))
113112a1dd 50 . . . . . . . . . . . 12 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → (¬ (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤))))))
114102, 113mt3d 140 . . . . . . . . . . 11 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)))
115108, 114syld 47 . . . . . . . . . 10 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → (¬ ⊥ → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)))
116 id 22 . . . . . . . . . . . . . . . . . 18 (¬ (¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V) ∨ ¬ 𝑦 ∈ V) → ¬ (¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V) ∨ ¬ 𝑦 ∈ V))
117116notornotel2 33527 . . . . . . . . . . . . . . . . 17 (¬ (¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V) ∨ ¬ 𝑦 ∈ V) → 𝑦 ∈ V)
118117a1i 11 . . . . . . . . . . . . . . . 16 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → (¬ (¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V) ∨ ¬ 𝑦 ∈ V) → 𝑦 ∈ V))
119116notornotel1 33526 . . . . . . . . . . . . . . . . . 18 (¬ (¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V) ∨ ¬ 𝑦 ∈ V) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V))
120119a1i 11 . . . . . . . . . . . . . . . . 17 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → (¬ (¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V) ∨ ¬ 𝑦 ∈ V) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)))
121 tsbi3 33571 . . . . . . . . . . . . . . . . . 18 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → ((𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ∨ ¬ 𝑦 ∈ V) ∨ ¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)))
122121a1d 25 . . . . . . . . . . . . . . . . 17 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → (¬ (¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V) ∨ ¬ 𝑦 ∈ V) → ((𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ∨ ¬ 𝑦 ∈ V) ∨ ¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V))))
123120, 122cnfn2dd 33524 . . . . . . . . . . . . . . . 16 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → (¬ (¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V) ∨ ¬ 𝑦 ∈ V) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ∨ ¬ 𝑦 ∈ V)))
124118, 123cnfn2dd 33524 . . . . . . . . . . . . . . 15 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → (¬ (¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V) ∨ ¬ 𝑦 ∈ V) → 𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V)))
125 a1tru 1497 . . . . . . . . . . . . . . . . 17 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → ⊤)
126125a1d 25 . . . . . . . . . . . . . . . 16 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → (¬ (¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V) ∨ ¬ 𝑦 ∈ V) → ⊤))
127 tsbi1 33569 . . . . . . . . . . . . . . . . . 18 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → ((¬ 𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ∨ ¬ ⊤) ∨ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))
128127a1d 25 . . . . . . . . . . . . . . . . 17 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → (¬ (¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V) ∨ ¬ 𝑦 ∈ V) → ((¬ 𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ∨ ¬ ⊤) ∨ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤))))
129128or32dd 33525 . . . . . . . . . . . . . . . 16 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → (¬ (¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V) ∨ ¬ 𝑦 ∈ V) → ((¬ 𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ∨ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) ∨ ¬ ⊤)))
130126, 129cnfn2dd 33524 . . . . . . . . . . . . . . 15 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → (¬ (¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V) ∨ ¬ 𝑦 ∈ V) → (¬ 𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ∨ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤))))
131124, 130cnfn1dd 33523 . . . . . . . . . . . . . 14 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → (¬ (¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V) ∨ ¬ 𝑦 ∈ V) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))
132131a1dd 50 . . . . . . . . . . . . 13 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → (¬ (¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V) ∨ ¬ 𝑦 ∈ V) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤))))
133132a1dd 50 . . . . . . . . . . . 12 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → (¬ (¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V) ∨ ¬ 𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))))
134 ax-1 6 . . . . . . . . . . . . 13 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → (¬ (¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V) ∨ ¬ 𝑦 ∈ V) → ¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤))))))
135 tsim3 33568 . . . . . . . . . . . . . 14 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤))) ∨ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤))))))
136135a1d 25 . . . . . . . . . . . . 13 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → (¬ (¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V) ∨ ¬ 𝑦 ∈ V) → (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤))) ∨ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))))))
137134, 136cnf2dd 33522 . . . . . . . . . . . 12 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → (¬ (¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V) ∨ ¬ 𝑦 ∈ V) → ¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))))
138133, 137contrd 33528 . . . . . . . . . . 11 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → (¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V) ∨ ¬ 𝑦 ∈ V))
139138a1d 25 . . . . . . . . . 10 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → (¬ ⊥ → (¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V) ∨ ¬ 𝑦 ∈ V)))
140115, 139cnfn1dd 33523 . . . . . . . . 9 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → (¬ ⊥ → ¬ 𝑦 ∈ V))
141110, 140contrd 33528 . . . . . . . 8 (¬ ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))) → ⊥)
142141efald2 33506 . . . . . . 7 ((¬ ∃𝑦𝜑𝑦 ∈ V) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤))))
143101, 142ax-mp 5 . . . . . 6 ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ 𝑦 ∈ V)) → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))
14410, 143ax-mp 5 . . . . 5 (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤))
145 ax-1 6 . . . . . . . . . 10 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → (¬ ⊥ → ¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))))
146 tsim3 33568 . . . . . . . . . . 11 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → (¬ (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))) ∨ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))))
147146a1d 25 . . . . . . . . . 10 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → (¬ ⊥ → (¬ (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))) ∨ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))))))
148145, 147cnf2dd 33522 . . . . . . . . 9 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → (¬ ⊥ → ¬ (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))))
149 tsim2 33567 . . . . . . . . . 10 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → (¬ ∃𝑦𝜑 ∨ (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))))
150149a1d 25 . . . . . . . . 9 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → (¬ ⊥ → (¬ ∃𝑦𝜑 ∨ (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))))
151148, 150cnf2dd 33522 . . . . . . . 8 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → (¬ ⊥ → ¬ ∃𝑦𝜑))
152 tsim2 33567 . . . . . . . . . 10 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) ∨ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))))
153152a1d 25 . . . . . . . . 9 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → (¬ ⊥ → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) ∨ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))))))
154145, 153cnf2dd 33522 . . . . . . . 8 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → (¬ ⊥ → (¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤))))
155151, 154mpdd 43 . . . . . . 7 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → (¬ ⊥ → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))
156 id 22 . . . . . . . . . . 11 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → ¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))))
157 id 22 . . . . . . . . . . . . . . 15 (¬ (∃𝑦𝜑𝜓) → ¬ (∃𝑦𝜑𝜓))
158157a1i 11 . . . . . . . . . . . . . 14 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → (¬ (∃𝑦𝜑𝜓) → ¬ (∃𝑦𝜑𝜓)))
159 tsim2 33567 . . . . . . . . . . . . . . 15 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → (∃𝑦𝜑 ∨ (∃𝑦𝜑𝜓)))
160159a1d 25 . . . . . . . . . . . . . 14 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → (¬ (∃𝑦𝜑𝜓) → (∃𝑦𝜑 ∨ (∃𝑦𝜑𝜓))))
161158, 160cnf2dd 33522 . . . . . . . . . . . . 13 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → (¬ (∃𝑦𝜑𝜓) → ∃𝑦𝜑))
162149a1d 25 . . . . . . . . . . . . 13 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → (¬ (∃𝑦𝜑𝜓) → (¬ ∃𝑦𝜑 ∨ (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))))
163161, 162cnfn1dd 33523 . . . . . . . . . . . 12 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → (¬ (∃𝑦𝜑𝜓) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))))
164163a1dd 50 . . . . . . . . . . 11 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → (¬ (∃𝑦𝜑𝜓) → ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))))
165156, 164mt3d 140 . . . . . . . . . 10 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → (∃𝑦𝜑𝜓))
166165a1d 25 . . . . . . . . 9 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → (¬ ⊥ → (∃𝑦𝜑𝜓)))
167 tsim3 33568 . . . . . . . . . . . . 13 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → (¬ (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))) ∨ (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))))
168167a1d 25 . . . . . . . . . . . 12 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → (¬ ⊥ → (¬ (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))) ∨ (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))))
169148, 168cnf2dd 33522 . . . . . . . . . . 11 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → (¬ ⊥ → ¬ (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))
170 tsim3 33568 . . . . . . . . . . . 12 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → (¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)) ∨ (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))
171170a1d 25 . . . . . . . . . . 11 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → (¬ ⊥ → (¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)) ∨ (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))))
172169, 171cnf2dd 33522 . . . . . . . . . 10 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → (¬ ⊥ → ¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))
173 tsbi1 33569 . . . . . . . . . . 11 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → ((¬ 𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ∨ ¬ (∃𝑦𝜑𝜓)) ∨ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))
174173a1d 25 . . . . . . . . . 10 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → (¬ ⊥ → ((¬ 𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ∨ ¬ (∃𝑦𝜑𝜓)) ∨ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))
175172, 174cnf2dd 33522 . . . . . . . . 9 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → (¬ ⊥ → (¬ 𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ∨ ¬ (∃𝑦𝜑𝜓))))
176166, 175cnfn2dd 33524 . . . . . . . 8 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → (¬ ⊥ → ¬ 𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V)))
177 a1tru 1497 . . . . . . . . . 10 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → ⊤)
178177a1d 25 . . . . . . . . 9 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → (¬ ⊥ → ⊤))
179 tsbi3 33571 . . . . . . . . . . 11 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → ((𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ∨ ¬ ⊤) ∨ ¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))
180179a1d 25 . . . . . . . . . 10 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → (¬ ⊥ → ((𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ∨ ¬ ⊤) ∨ ¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤))))
181180or32dd 33525 . . . . . . . . 9 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → (¬ ⊥ → ((𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ∨ ¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) ∨ ¬ ⊤)))
182178, 181cnfn2dd 33524 . . . . . . . 8 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → (¬ ⊥ → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ∨ ¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤))))
183176, 182cnf1dd 33521 . . . . . . 7 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → (¬ ⊥ → ¬ (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)))
184155, 183contrd 33528 . . . . . 6 (¬ ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))) → ⊥)
185184efald2 33506 . . . . 5 ((¬ ∃𝑦𝜑 → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ ⊤)) → (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))))
186144, 185ax-mp 5 . . . 4 (¬ ∃𝑦𝜑 → (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓))))
187100, 186pm2.61i 176 . . 3 (𝑦 = (𝑓𝑥) → (𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) ↔ (∃𝑦𝜑𝜓)))
18817, 18, 187ac6s3f 33608 . 2 (∀𝑥𝐴𝑦 𝑦 ∈ if(∃𝑦𝜑, {𝑦𝜑}, V) → ∃𝑓𝑥𝐴 (∃𝑦𝜑𝜓))
18914, 188ax-mp 5 1 𝑓𝑥𝐴 (∃𝑦𝜑𝜓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   = wceq 1480  ⊤wtru 1481  ⊥wfal 1485  ∃wex 1701  Ⅎwnf 1705   ∈ wcel 1987  {cab 2607  ∀wral 2907  Vcvv 3186  ifcif 4058  ‘cfv 5847 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-reg 8441  ax-inf2 8482  ax-ac2 9229 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-en 7900  df-r1 8571  df-rank 8572  df-card 8709  df-ac 8883 This theorem is referenced by:  ac6s6f  33610
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