Theorem vtoclgft 3510

Description: Closed theorem form of vtoclgf 3526. The reverse implication is proven in ceqsal1t 3474. See ceqsalt 3475 for a version with 𝑥 and 𝐴 disjoint. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by JJ, 11-Aug-2021.) Avoid ax-13 2377. (Revised by GG, 6-Oct-2023.)

Assertion

Ref	Expression
vtoclgft	⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ 𝑉) → 𝜓)

Proof of Theorem vtoclgft

Step	Hyp	Ref	Expression
1		biimp 215	. . . . . . 7 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
2	1	imim2i 16	. . . . . 6 ⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝑥 = 𝐴 → (𝜑 → 𝜓)))
3	2	alimi 1813	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → ∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)))
4		spcimgft 3504	. . . . 5 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓))) → (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓)))
5	3, 4	sylan2 594	. . . 4 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓)))
6	5	com23 86	. . 3 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (∀𝑥𝜑 → (𝐴 ∈ 𝑉 → 𝜓)))
7	6	impr 454	. 2 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥𝜑)) → (𝐴 ∈ 𝑉 → 𝜓))
8	7	3impia 1118	1 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ 𝑉) → 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087  ∀wal 1540   = wceq 1542  Ⅎwnf 1785   ∈ wcel 2114  Ⅎwnfc 2884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-ex 1782  df-nf 1786  df-cleq 2729  df-clel 2812  df-nfc 2886

This theorem is referenced by:  vtocldf  3518  bj-vtoclgfALT  37262