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Theorem equsexd 1658
Description: Deduction form of equsex 1657. (Contributed by Jim Kingdon, 29-Dec-2017.)
Hypotheses
Ref Expression
equsexd.1 (𝜑 → ∀𝑥𝜑)
equsexd.2 (𝜑 → (𝜒 → ∀𝑥𝜒))
equsexd.3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
equsexd (𝜑 → (∃𝑥(𝑥 = 𝑦𝜓) ↔ 𝜒))

Proof of Theorem equsexd
StepHypRef Expression
1 equsexd.1 . . 3 (𝜑 → ∀𝑥𝜑)
2 equsexd.2 . . 3 (𝜑 → (𝜒 → ∀𝑥𝜒))
3 equsexd.3 . . . 4 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
4 bi1 116 . . . . 5 ((𝜓𝜒) → (𝜓𝜒))
54imim2i 12 . . . 4 ((𝑥 = 𝑦 → (𝜓𝜒)) → (𝑥 = 𝑦 → (𝜓𝜒)))
6 pm3.31 258 . . . 4 ((𝑥 = 𝑦 → (𝜓𝜒)) → ((𝑥 = 𝑦𝜓) → 𝜒))
73, 5, 63syl 17 . . 3 (𝜑 → ((𝑥 = 𝑦𝜓) → 𝜒))
81, 2, 7exlimd2 1527 . 2 (𝜑 → (∃𝑥(𝑥 = 𝑦𝜓) → 𝜒))
9 a9e 1627 . . . 4 𝑥 𝑥 = 𝑦
101a1i 9 . . . . . . . . 9 (𝜑 → (𝜑 → ∀𝑥𝜑))
1110, 2jca 300 . . . . . . . 8 (𝜑 → ((𝜑 → ∀𝑥𝜑) ∧ (𝜒 → ∀𝑥𝜒)))
12 prth 336 . . . . . . . 8 (((𝜑 → ∀𝑥𝜑) ∧ (𝜒 → ∀𝑥𝜒)) → ((𝜑𝜒) → (∀𝑥𝜑 ∧ ∀𝑥𝜒)))
1311, 12syl 14 . . . . . . 7 (𝜑 → ((𝜑𝜒) → (∀𝑥𝜑 ∧ ∀𝑥𝜒)))
14 19.26 1411 . . . . . . 7 (∀𝑥(𝜑𝜒) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜒))
1513, 14syl6ibr 160 . . . . . 6 (𝜑 → ((𝜑𝜒) → ∀𝑥(𝜑𝜒)))
1615anabsi5 544 . . . . 5 ((𝜑𝜒) → ∀𝑥(𝜑𝜒))
17 idd 21 . . . . . . . 8 (𝜒 → (𝑥 = 𝑦𝑥 = 𝑦))
1817a1i 9 . . . . . . 7 (𝜑 → (𝜒 → (𝑥 = 𝑦𝑥 = 𝑦)))
1918imp 122 . . . . . 6 ((𝜑𝜒) → (𝑥 = 𝑦𝑥 = 𝑦))
20 bi2 128 . . . . . . . . 9 ((𝜓𝜒) → (𝜒𝜓))
2120imim2i 12 . . . . . . . 8 ((𝑥 = 𝑦 → (𝜓𝜒)) → (𝑥 = 𝑦 → (𝜒𝜓)))
22 pm2.04 81 . . . . . . . 8 ((𝑥 = 𝑦 → (𝜒𝜓)) → (𝜒 → (𝑥 = 𝑦𝜓)))
233, 21, 223syl 17 . . . . . . 7 (𝜑 → (𝜒 → (𝑥 = 𝑦𝜓)))
2423imp 122 . . . . . 6 ((𝜑𝜒) → (𝑥 = 𝑦𝜓))
2519, 24jcad 301 . . . . 5 ((𝜑𝜒) → (𝑥 = 𝑦 → (𝑥 = 𝑦𝜓)))
2616, 25eximdh 1543 . . . 4 ((𝜑𝜒) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥(𝑥 = 𝑦𝜓)))
279, 26mpi 15 . . 3 ((𝜑𝜒) → ∃𝑥(𝑥 = 𝑦𝜓))
2827ex 113 . 2 (𝜑 → (𝜒 → ∃𝑥(𝑥 = 𝑦𝜓)))
298, 28impbid 127 1 (𝜑 → (∃𝑥(𝑥 = 𝑦𝜓) ↔ 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1283   = wceq 1285  wex 1422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  cbvexdh  1843
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