Theorem ceqsalt 2707

Description: Closed theorem version of ceqsalg 2709. (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		elisset 2695	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
2	1	3ad2ant3 1004	. . 3 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → ∃𝑥 𝑥 = 𝐴)
3		bi1 117	. . . . . . 7 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
4	3	imim3i 61	. . . . . 6 ⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → ((𝑥 = 𝐴 → 𝜑) → (𝑥 = 𝐴 → 𝜓)))
5	4	al2imi 1434	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (∀𝑥(𝑥 = 𝐴 → 𝜑) → ∀𝑥(𝑥 = 𝐴 → 𝜓)))
6	5	3ad2ant2 1003	. . . 4 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) → ∀𝑥(𝑥 = 𝐴 → 𝜓)))
7		19.23t 1655	. . . . 5 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ (∃𝑥 𝑥 = 𝐴 → 𝜓)))
8	7	3ad2ant1 1002	. . . 4 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ (∃𝑥 𝑥 = 𝐴 → 𝜓)))
9	6, 8	sylibd 148	. . 3 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) → (∃𝑥 𝑥 = 𝐴 → 𝜓)))
10	2, 9	mpid 42	. 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) → 𝜓))
11		bi2 129	. . . . . . 7 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
12	11	imim2i 12	. . . . . 6 ⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝑥 = 𝐴 → (𝜓 → 𝜑)))
13	12	com23 78	. . . . 5 ⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝜓 → (𝑥 = 𝐴 → 𝜑)))
14	13	alimi 1431	. . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → ∀𝑥(𝜓 → (𝑥 = 𝐴 → 𝜑)))
15	14	3ad2ant2 1003	. . 3 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → ∀𝑥(𝜓 → (𝑥 = 𝐴 → 𝜑)))
16		19.21t 1561	. . . 4 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜓 → (𝑥 = 𝐴 → 𝜑)) ↔ (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑))))
17	16	3ad2ant1 1002	. . 3 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝜓 → (𝑥 = 𝐴 → 𝜑)) ↔ (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑))))
18	15, 17	mpbid 146	. 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑)))
19	10, 18	impbid 128	1 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 104   ∧ w3a 962  ∀wal 1329   = wceq 1331  Ⅎwnf 1436  ∃wex 1468   ∈ wcel 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-v 2683

This theorem is referenced by:  ceqsralt  2708