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Theorem bj-vtoclgft 14409
Description: Weakening two hypotheses of vtoclgf 2795. (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 2748 . 2 (𝐴𝑉𝐴 ∈ V)
2 bj-vtoclgf.nf1 . . . 4 𝑥𝐴
32issetf 2744 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
4 bj-vtoclgf.nf2 . . . 4 𝑥𝜓
5 bj-vtoclgf.min . . . 4 (𝑥 = 𝐴𝜑)
64, 5bj-exlimmp 14403 . . 3 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (∃𝑥 𝑥 = 𝐴𝜓))
73, 6biimtrid 152 . 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 1351   = wceq 1353  wnf 1460  wex 1492  wcel 2148  wnfc 2306  Vcvv 2737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739
This theorem is referenced by:  bj-vtoclgf  14410  elabgft1  14412  bj-rspgt  14420
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