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Theorem vtoclgaf 3422
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 (𝑥𝐵𝜑)
Assertion
Ref Expression
vtoclgaf (𝐴𝐵𝜓)
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 2928 . . . 4 𝑥 𝐴𝐵
3 vtoclgaf.2 . . . 4 𝑥𝜓
42, 3nfim 1977 . . 3 𝑥(𝐴𝐵𝜓)
5 eleq1 2838 . . . 4 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
6 vtoclgaf.3 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
75, 6imbi12d 333 . . 3 (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ (𝐴𝐵𝜓)))
8 vtoclgaf.4 . . 3 (𝑥𝐵𝜑)
91, 4, 7, 8vtoclgf 3415 . 2 (𝐴𝐵 → (𝐴𝐵𝜓))
109pm2.43i 52 1 (𝐴𝐵𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1631  wnf 1856  wcel 2145  wnfc 2900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353
This theorem is referenced by:  vtoclga  3423  ssiun2s  4699  iunopeqop  5115  fvmptss  6436  fvmptf  6445  fmptco  6542  tfis  7205  inar1  9803  sumss  14663  fprodn0  14916  prmind2  15605  lss1d  19176  itg2splitlem  23735  dgrle  24219  cnlnadjlem5  29270  poimirlem25  33766  stoweidlem26  40755
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