Theorem bj-vtoclgft 16139

Description: Weakening two hypotheses of vtoclgf 2859. (Contributed by BJ, 21-Nov-2019.)

Hypotheses

Ref	Expression
bj-vtoclgf.nf1	⊢ Ⅎ𝑥𝐴
bj-vtoclgf.nf2	⊢ Ⅎ𝑥𝜓
bj-vtoclgf.min	⊢ (𝑥 = 𝐴 → 𝜑)

Assertion

Ref	Expression
bj-vtoclgft	⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝑉 → 𝜓))

Proof of Theorem bj-vtoclgft

Step	Hyp	Ref	Expression
1		elex 2811	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		bj-vtoclgf.nf1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	2	issetf 2807	. . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
4		bj-vtoclgf.nf2	. . . 4 ⊢ Ⅎ𝑥𝜓
5		bj-vtoclgf.min	. . . 4 ⊢ (𝑥 = 𝐴 → 𝜑)
6	4, 5	bj-exlimmp 16133	. . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (∃𝑥 𝑥 = 𝐴 → 𝜓))
7	3, 6	biimtrid 152	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ V → 𝜓))
8	1, 7	syl5 32	1 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝑉 → 𝜓))

Syntax hints:   → wi 4  ∀wal 1393   = wceq 1395  Ⅎwnf 1506  ∃wex 1538   ∈ wcel 2200  Ⅎwnfc 2359  Vcvv 2799

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801

This theorem is referenced by:  bj-vtoclgf  16140  elabgft1  16142  bj-rspgt  16150