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Theorem bj-vtoclgf 12910
Description: Weakening two hypotheses of vtoclgf 2718. (Contributed by BJ, 21-Nov-2019.)
Hypotheses
Ref Expression
bj-vtoclgf.nf1 𝑥𝐴
bj-vtoclgf.nf2 𝑥𝜓
bj-vtoclgf.min (𝑥 = 𝐴𝜑)
bj-vtoclgf.maj (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
bj-vtoclgf (𝐴𝑉𝜓)

Proof of Theorem bj-vtoclgf
StepHypRef Expression
1 bj-vtoclgf.nf1 . . 3 𝑥𝐴
2 bj-vtoclgf.nf2 . . 3 𝑥𝜓
3 bj-vtoclgf.min . . 3 (𝑥 = 𝐴𝜑)
41, 2, 3bj-vtoclgft 12909 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝑉𝜓))
5 bj-vtoclgf.maj . 2 (𝑥 = 𝐴 → (𝜑𝜓))
64, 5mpg 1412 1 (𝐴𝑉𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316  wnf 1421  wcel 1465  wnfc 2245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662
This theorem is referenced by:  elabgf2  12914
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