Theorem bj-vtoclgft 15911

Description: Weakening two hypotheses of vtoclgf 2836. (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 2788	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		bj-vtoclgf.nf1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	2	issetf 2784	. . 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 15905	. . 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 1371   = wceq 1373  Ⅎwnf 1484  ∃wex 1516   ∈ wcel 2178  Ⅎwnfc 2337  Vcvv 2776

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778

This theorem is referenced by:  bj-vtoclgf  15912  elabgft1  15914  bj-rspgt  15922