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Theorem vtoclbg 2718
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 29-Apr-1994.)
Hypotheses
Ref Expression
vtoclbg.1 (𝑥 = 𝐴 → (𝜑𝜒))
vtoclbg.2 (𝑥 = 𝐴 → (𝜓𝜃))
vtoclbg.3 (𝜑𝜓)
Assertion
Ref Expression
vtoclbg (𝐴𝑉 → (𝜒𝜃))
Distinct variable groups:   𝑥,𝐴   𝜒,𝑥   𝜃,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem vtoclbg
StepHypRef Expression
1 vtoclbg.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜒))
2 vtoclbg.2 . . 3 (𝑥 = 𝐴 → (𝜓𝜃))
31, 2bibi12d 234 . 2 (𝑥 = 𝐴 → ((𝜑𝜓) ↔ (𝜒𝜃)))
4 vtoclbg.3 . 2 (𝜑𝜓)
53, 4vtoclg 2717 1 (𝐴𝑉 → (𝜒𝜃))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1314  wcel 1463
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659
This theorem is referenced by:  pm13.183  2792  sbc8g  2885  sbcco  2899  sbc5  2901  sbcie2g  2910  eqsbc3  2916  sbcng  2917  sbcimg  2918  sbcan  2919  sbcang  2920  sbcor  2921  sbcorg  2922  sbcbig  2923  sbcal  2928  sbcalg  2929  sbcex2  2930  sbcexg  2931  sbcel1v  2939  sbcralg  2955  sbcreug  2957  sbcel12g  2984  sbceqg  2985  csbiebg  3008  elpwg  3484  snssg  3622  preq12bg  3666  elintg  3745  elintrabg  3750  sbcbrg  3944  opelresg  4784  elixpsn  6583  ixpsnf1o  6584  domeng  6600
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