Theorem ceqsalt 3515

Description: Closed theorem version of ceqsalg 3517. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem ceqsalt

Step	Hyp	Ref	Expression
1		biimp 215	. . . . . . 7 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
2	1	imim3i 64	. . . . . 6 ⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → ((𝑥 = 𝐴 → 𝜑) → (𝑥 = 𝐴 → 𝜓)))
3	2	al2imi 1815	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (∀𝑥(𝑥 = 𝐴 → 𝜑) → ∀𝑥(𝑥 = 𝐴 → 𝜓)))
4		elisset 2823	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
5		19.23t 2210	. . . . . . 7 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ (∃𝑥 𝑥 = 𝐴 → 𝜓)))
6	5	biimpd 229	. . . . . 6 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝑥 = 𝐴 → 𝜓) → (∃𝑥 𝑥 = 𝐴 → 𝜓)))
7	4, 6	syl7 74	. . . . 5 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝑥 = 𝐴 → 𝜓) → (𝐴 ∈ 𝑉 → 𝜓)))
8	3, 7	sylan9r 508	. . . 4 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (∀𝑥(𝑥 = 𝐴 → 𝜑) → (𝐴 ∈ 𝑉 → 𝜓)))
9	8	com23 86	. . 3 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) → 𝜓)))
10	9	3impia 1118	. 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) → 𝜓))
11		ceqsal1t 3514	. . 3 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑)))
12	11	3adant3 1133	. 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑)))
13	10, 12	impbid 212	1 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087  ∀wal 1538   = wceq 1540  ∃wex 1779  Ⅎwnf 1783   ∈ wcel 2108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-12 2177

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-clel 2816

This theorem is referenced by:  ceqsralt  3516  ceqsalg  3517  vtoclegft  3588  sbciegft  3825