Theorem bj-vtoclgft 16374

Description: Weakening two hypotheses of vtoclgf 2862. (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 2814	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		bj-vtoclgf.nf1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	2	issetf 2810	. . 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 16368	. . 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 1395   = wceq 1397  Ⅎwnf 1508  ∃wex 1540   ∈ wcel 2202  Ⅎwnfc 2361  Vcvv 2802

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804

This theorem is referenced by:  bj-vtoclgf  16375  elabgft1  16377  bj-rspgt  16385