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Theorem bj-vtoclgft 13041
Description: Weakening two hypotheses of vtoclgf 2744. (Contributed by BJ, 21-Nov-2019.)
Hypotheses
Ref Expression
bj-vtoclgf.nf1 𝑥𝐴
bj-vtoclgf.nf2 𝑥𝜓
bj-vtoclgf.min (𝑥 = 𝐴𝜑)
Assertion
Ref Expression
bj-vtoclgft (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝑉𝜓))

Proof of Theorem bj-vtoclgft
StepHypRef Expression
1 elex 2697 . 2 (𝐴𝑉𝐴 ∈ V)
2 bj-vtoclgf.nf1 . . . 4 𝑥𝐴
32issetf 2693 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
4 bj-vtoclgf.nf2 . . . 4 𝑥𝜓
5 bj-vtoclgf.min . . . 4 (𝑥 = 𝐴𝜑)
64, 5bj-exlimmp 13035 . . 3 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (∃𝑥 𝑥 = 𝐴𝜓))
73, 6syl5bi 151 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴 ∈ V → 𝜓))
81, 7syl5 32 1 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝑉𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1329   = wceq 1331  wnf 1436  wex 1468  wcel 1480  wnfc 2268  Vcvv 2686
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688
This theorem is referenced by:  bj-vtoclgf  13042  elabgft1  13044  bj-rspgt  13052
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