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Theorem vtoclbg 2608
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 (A 𝑉 → (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 224 . 2 (x = A → ((φψ) ↔ (χθ)))
4 vtoclbg.3 . 2 (φψ)
53, 4vtoclg 2607 1 (A 𝑉 → (χθ))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242   wcel 1390
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-bndl 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:  pm13.183  2675  sbc8g  2765  sbcco  2779  sbc5  2781  sbcie2g  2790  eqsbc3  2796  sbcng  2797  sbcimg  2798  sbcan  2799  sbcang  2800  sbcor  2801  sbcorg  2802  sbcbig  2803  sbcal  2804  sbcalg  2805  sbcex2  2806  sbcexg  2807  sbc3ang  2814  sbcel1gv  2815  sbcralg  2830  sbcrexg  2831  sbcreug  2832  sbcel12g  2859  sbceqg  2860  csbiebg  2883  elpwg  3359  snssg  3491  preq12bg  3535  elintg  3614  elintrabg  3619  sbcbrg  3804  opelresg  4562  domeng  6169
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