Theorem vtoclgft 3564

Description: Closed theorem form of vtoclgf 3581. The reverse implication is proven in ceqsal1t 3522. See ceqsalt 3523 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 2380. (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 1809	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → ∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)))
4		spcimgft 3558	. . . . 5 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓))) → (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓)))
5	3, 4	sylan2 592	. . . 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 1117	1 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ 𝑉) → 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087  ∀wal 1535   = wceq 1537  Ⅎwnf 1781   ∈ wcel 2108  Ⅎwnfc 2893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-ex 1778  df-nf 1782  df-cleq 2732  df-clel 2819  df-nfc 2895

This theorem is referenced by:  vtocldf  3572  bj-vtoclgfALT  37025