Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-vtoclgft GIF version

Theorem bj-vtoclgft 15212
Description: Weakening two hypotheses of vtoclgf 2818. (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 2771 . 2 (𝐴𝑉𝐴 ∈ V)
2 bj-vtoclgf.nf1 . . . 4 𝑥𝐴
32issetf 2767 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
4 bj-vtoclgf.nf2 . . . 4 𝑥𝜓
5 bj-vtoclgf.min . . . 4 (𝑥 = 𝐴𝜑)
64, 5bj-exlimmp 15206 . . 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 1362   = wceq 1364  wnf 1471  wex 1503  wcel 2164  wnfc 2323  Vcvv 2760
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762
This theorem is referenced by:  bj-vtoclgf  15213  elabgft1  15215  bj-rspgt  15223
  Copyright terms: Public domain W3C validator