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Theorem bj-vtoclgft 9914
 Description: Weakening two hypotheses of vtoclgf 2612. (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 2566 . 2 (𝐴𝑉𝐴 ∈ V)
2 bj-vtoclgf.nf1 . . . 4 𝑥𝐴
32issetf 2562 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
4 bj-vtoclgf.nf2 . . . 4 𝑥𝜓
5 bj-vtoclgf.min . . . 4 (𝑥 = 𝐴𝜑)
64, 5bj-exlimmp 9909 . . 3 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (∃𝑥 𝑥 = 𝐴𝜓))
73, 6syl5bi 141 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴 ∈ V → 𝜓))
81, 7syl5 28 1 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝑉𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1241   = wceq 1243  Ⅎwnf 1349  ∃wex 1381   ∈ wcel 1393  Ⅎwnfc 2165  Vcvv 2557 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559 This theorem is referenced by:  bj-vtoclgf  9915  elabgft1  9917  bj-rspgt  9925
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