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Theorem vtoclbg 2631
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 228 . 2 (𝑥 = 𝐴 → ((𝜑𝜓) ↔ (𝜒𝜃)))
4 vtoclbg.3 . 2 (𝜑𝜓)
53, 4vtoclg 2630 1 (𝐴𝑉 → (𝜒𝜃))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102   = wceq 1259  wcel 1409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576
This theorem is referenced by:  pm13.183  2704  sbc8g  2794  sbcco  2808  sbc5  2810  sbcie2g  2819  eqsbc3  2825  sbcng  2826  sbcimg  2827  sbcan  2828  sbcang  2829  sbcor  2830  sbcorg  2831  sbcbig  2832  sbcal  2837  sbcalg  2838  sbcex2  2839  sbcexg  2840  sbcel1v  2848  sbcralg  2864  sbcreug  2866  sbcel12g  2893  sbceqg  2894  csbiebg  2917  elpwg  3395  snssg  3528  preq12bg  3572  elintg  3651  elintrabg  3656  sbcbrg  3841  opelresg  4647  domeng  6264
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