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Theorem csbunigVD 44896
Description: Virtual deduction proof of csbuni 4941. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. csbuni 4941 is csbunigVD 44896 without virtual deductions and was automatically derived from csbunigVD 44896.
1:: (   𝐴𝑉   ▶   𝐴𝑉   )
2:1: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑧𝑦𝑧 𝑦)   )
3:1: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦𝐵𝑦 𝐴 / 𝑥𝐵)   )
4:2,3: (   𝐴𝑉   ▶   (([𝐴 / 𝑥]𝑧𝑦 [𝐴 / 𝑥]𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵))   )
5:1: (   𝐴𝑉   ▶   ([𝐴 / 𝑥](𝑧𝑦 𝑦𝐵) ↔ ([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵))   )
6:4,5: (   𝐴𝑉   ▶   ([𝐴 / 𝑥](𝑧𝑦 𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵))   )
7:6: (   𝐴𝑉   ▶   𝑦([𝐴 / 𝑥](𝑧 𝑦𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵))   )
8:7: (   𝐴𝑉   ▶   (∃𝑦[𝐴 / 𝑥](𝑧 𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵))   )
9:1: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦(𝑧 𝑦𝑦𝐵) ↔ ∃𝑦[𝐴 / 𝑥](𝑧𝑦𝑦𝐵))   )
10:8,9: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦(𝑧 𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵))   )
11:10: (   𝐴𝑉   ▶   𝑧([𝐴 / 𝑥]𝑦( 𝑧𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵))   )
12:11: (   𝐴𝑉   ▶   {𝑧[𝐴 / 𝑥]𝑦( 𝑧𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦 𝑦𝐴 / 𝑥𝐵)}   )
13:1: (   𝐴𝑉   ▶   𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧 𝑦𝑦𝐵)} = {𝑧[𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵)}    )
14:12,13: (   𝐴𝑉   ▶   𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧 𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦 𝑦𝐴 / 𝑥𝐵)}   )
15:: 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)}
16:15: 𝑥 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦 𝐵)}
17:1,16: (   𝐴𝑉   ▶   [𝐴 / 𝑥] 𝐵 = {𝑧 𝑦(𝑧𝑦𝑦𝐵)}   )
18:1,17: (   𝐴𝑉   ▶   𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)}   )
19:14,18: (   𝐴𝑉   ▶   𝐴 / 𝑥 𝐵 = {𝑧 𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)}   )
20:: 𝐴 / 𝑥𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦 𝑦𝐴 / 𝑥𝐵)}
21:19,20: (   𝐴𝑉   ▶   𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵   )
qed:21: (𝐴𝑉𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵)
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
csbunigVD (𝐴𝑉𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵)

Proof of Theorem csbunigVD
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idn1 44572 . . . . . . . . . . . . 13 (   𝐴𝑉   ▶   𝐴𝑉   )
2 sbcg 3870 . . . . . . . . . . . . 13 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝑦𝑧𝑦))
31, 2e1a 44625 . . . . . . . . . . . 12 (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑧𝑦𝑧𝑦)   )
4 sbcel2 4424 . . . . . . . . . . . . . 14 ([𝐴 / 𝑥]𝑦𝐵𝑦𝐴 / 𝑥𝐵)
54a1i 11 . . . . . . . . . . . . 13 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐵𝑦𝐴 / 𝑥𝐵))
61, 5e1a 44625 . . . . . . . . . . . 12 (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦𝐵𝑦𝐴 / 𝑥𝐵)   )
7 pm4.38 637 . . . . . . . . . . . . 13 ((([𝐴 / 𝑥]𝑧𝑦𝑧𝑦) ∧ ([𝐴 / 𝑥]𝑦𝐵𝑦𝐴 / 𝑥𝐵)) → (([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵)))
87ex 412 . . . . . . . . . . . 12 (([𝐴 / 𝑥]𝑧𝑦𝑧𝑦) → (([𝐴 / 𝑥]𝑦𝐵𝑦𝐴 / 𝑥𝐵) → (([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵))))
93, 6, 8e11 44686 . . . . . . . . . . 11 (   𝐴𝑉   ▶   (([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵))   )
10 sbcan 3844 . . . . . . . . . . . . 13 ([𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ ([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵))
1110a1i 11 . . . . . . . . . . . 12 (𝐴𝑉 → ([𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ ([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵)))
121, 11e1a 44625 . . . . . . . . . . 11 (   𝐴𝑉   ▶   ([𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ ([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵))   )
13 bibi1 351 . . . . . . . . . . . 12 (([𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ ([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵)) → (([𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵)) ↔ (([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵))))
1413biimprcd 250 . . . . . . . . . . 11 ((([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵)) → (([𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ ([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵)) → ([𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵))))
159, 12, 14e11 44686 . . . . . . . . . 10 (   𝐴𝑉   ▶   ([𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵))   )
1615gen11 44614 . . . . . . . . 9 (   𝐴𝑉   ▶   𝑦([𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵))   )
17 exbi 1844 . . . . . . . . 9 (∀𝑦([𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵)) → (∃𝑦[𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)))
1816, 17e1a 44625 . . . . . . . 8 (   𝐴𝑉   ▶   (∃𝑦[𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵))   )
19 sbcex2 3856 . . . . . . . . . 10 ([𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵) ↔ ∃𝑦[𝐴 / 𝑥](𝑧𝑦𝑦𝐵))
2019a1i 11 . . . . . . . . 9 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵) ↔ ∃𝑦[𝐴 / 𝑥](𝑧𝑦𝑦𝐵)))
211, 20e1a 44625 . . . . . . . 8 (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵) ↔ ∃𝑦[𝐴 / 𝑥](𝑧𝑦𝑦𝐵))   )
22 bibi1 351 . . . . . . . . 9 (([𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵) ↔ ∃𝑦[𝐴 / 𝑥](𝑧𝑦𝑦𝐵)) → (([𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)) ↔ (∃𝑦[𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵))))
2322biimprcd 250 . . . . . . . 8 ((∃𝑦[𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)) → (([𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵) ↔ ∃𝑦[𝐴 / 𝑥](𝑧𝑦𝑦𝐵)) → ([𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵))))
2418, 21, 23e11 44686 . . . . . . 7 (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵))   )
2524gen11 44614 . . . . . 6 (   𝐴𝑉   ▶   𝑧([𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵))   )
26 abbib 2809 . . . . . . 7 ({𝑧[𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)} ↔ ∀𝑧([𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)))
2726biimpri 228 . . . . . 6 (∀𝑧([𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)) → {𝑧[𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)})
2825, 27e1a 44625 . . . . 5 (   𝐴𝑉   ▶   {𝑧[𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)}   )
29 csbab 4446 . . . . . . 7 𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} = {𝑧[𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵)}
3029a1i 11 . . . . . 6 (𝐴𝑉𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} = {𝑧[𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵)})
311, 30e1a 44625 . . . . 5 (   𝐴𝑉   ▶   𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} = {𝑧[𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵)}   )
32 eqeq2 2747 . . . . . 6 ({𝑧[𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)} → (𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} = {𝑧[𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵)} ↔ 𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)}))
3332biimpd 229 . . . . 5 ({𝑧[𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)} → (𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} = {𝑧[𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵)} → 𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)}))
3428, 31, 33e11 44686 . . . 4 (   𝐴𝑉   ▶   𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)}   )
35 df-uni 4913 . . . . . . 7 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)}
3635ax-gen 1792 . . . . . 6 𝑥 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)}
37 spsbc 3804 . . . . . 6 (𝐴𝑉 → (∀𝑥 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} → [𝐴 / 𝑥] 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)}))
381, 36, 37e10 44692 . . . . 5 (   𝐴𝑉   ▶   [𝐴 / 𝑥] 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)}   )
39 sbceqg 4418 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥] 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} ↔ 𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)}))
4039biimpd 229 . . . . 5 (𝐴𝑉 → ([𝐴 / 𝑥] 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} → 𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)}))
411, 38, 40e11 44686 . . . 4 (   𝐴𝑉   ▶   𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)}   )
42 eqeq2 2747 . . . . 5 (𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)} → (𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} ↔ 𝐴 / 𝑥 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)}))
4342biimpd 229 . . . 4 (𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)} → (𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} → 𝐴 / 𝑥 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)}))
4434, 41, 43e11 44686 . . 3 (   𝐴𝑉   ▶   𝐴 / 𝑥 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)}   )
45 df-uni 4913 . . 3 𝐴 / 𝑥𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)}
46 eqeq2 2747 . . . 4 ( 𝐴 / 𝑥𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)} → (𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵𝐴 / 𝑥 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)}))
4746biimprcd 250 . . 3 (𝐴 / 𝑥 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)} → ( 𝐴 / 𝑥𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)} → 𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵))
4844, 45, 47e10 44692 . 2 (   𝐴𝑉   ▶   𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵   )
4948in1 44569 1 (𝐴𝑉𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wex 1776  wcel 2106  {cab 2712  [wsbc 3791  csb 3908   cuni 4912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-nul 4340  df-uni 4913  df-vd1 44568
This theorem is referenced by: (None)
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