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Theorem vtoclgaf 2612
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 17-Feb-2006.) (Revised by Mario Carneiro, 10-Oct-2016.)
Hypotheses
Ref Expression
vtoclgaf.1 xA
vtoclgaf.2 xψ
vtoclgaf.3 (x = A → (φψ))
vtoclgaf.4 (x Bφ)
Assertion
Ref Expression
vtoclgaf (A Bψ)
Distinct variable group:   x,B
Allowed substitution hints:   φ(x)   ψ(x)   A(x)

Proof of Theorem vtoclgaf
StepHypRef Expression
1 vtoclgaf.1 . . 3 xA
21nfel1 2185 . . . 4 x A B
3 vtoclgaf.2 . . . 4 xψ
42, 3nfim 1461 . . 3 x(A Bψ)
5 eleq1 2097 . . . 4 (x = A → (x BA B))
6 vtoclgaf.3 . . . 4 (x = A → (φψ))
75, 6imbi12d 223 . . 3 (x = A → ((x Bφ) ↔ (A Bψ)))
8 vtoclgaf.4 . . 3 (x Bφ)
91, 4, 7, 8vtoclgf 2606 . 2 (A B → (A Bψ))
109pm2.43i 43 1 (A Bψ)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242  wnf 1346   wcel 1390  wnfc 2162
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553
This theorem is referenced by:  vtoclga  2613  ssiun2s  3692  tfis  4249  fvmptf  5206  fmptco  5273
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