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Theorem bj-vtoclgft 10301
Description: Weakening two hypotheses of vtoclgf 2629. (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 2583 . 2 (𝐴𝑉𝐴 ∈ V)
2 bj-vtoclgf.nf1 . . . 4 𝑥𝐴
32issetf 2579 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
4 bj-vtoclgf.nf2 . . . 4 𝑥𝜓
5 bj-vtoclgf.min . . . 4 (𝑥 = 𝐴𝜑)
64, 5bj-exlimmp 10296 . . 3 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (∃𝑥 𝑥 = 𝐴𝜓))
73, 6syl5bi 145 . 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 1257   = wceq 1259  wnf 1365  wex 1397  wcel 1409  wnfc 2181  Vcvv 2574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576
This theorem is referenced by:  bj-vtoclgf  10302  elabgft1  10304  bj-rspgt  10312
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