Theorem bj-vtoclgft 15272

Description: Weakening two hypotheses of vtoclgf 2818. (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 2771	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		bj-vtoclgf.nf1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	2	issetf 2767	. . 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 15266	. . 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 1362   = wceq 1364  Ⅎwnf 1471  ∃wex 1503   ∈ wcel 2164  Ⅎwnfc 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  15273  elabgft1  15275  bj-rspgt  15283