Theorem ceqsalt 2580

Description: Closed theorem version of ceqsalg 2582. (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 2568	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
2	1	3ad2ant3 927	. . 3 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → ∃𝑥 𝑥 = 𝐴)
3		bi1 111	. . . . . . 7 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
4	3	imim3i 55	. . . . . 6 ⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → ((𝑥 = 𝐴 → 𝜑) → (𝑥 = 𝐴 → 𝜓)))
5	4	al2imi 1347	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (∀𝑥(𝑥 = 𝐴 → 𝜑) → ∀𝑥(𝑥 = 𝐴 → 𝜓)))
6	5	3ad2ant2 926	. . . 4 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) → ∀𝑥(𝑥 = 𝐴 → 𝜓)))
7		19.23t 1567	. . . . 5 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ (∃𝑥 𝑥 = 𝐴 → 𝜓)))
8	7	3ad2ant1 925	. . . 4 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ (∃𝑥 𝑥 = 𝐴 → 𝜓)))
9	6, 8	sylibd 138	. . 3 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) → (∃𝑥 𝑥 = 𝐴 → 𝜓)))
10	2, 9	mpid 37	. 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) → 𝜓))
11		bi2 121	. . . . . . 7 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
12	11	imim2i 12	. . . . . 6 ⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝑥 = 𝐴 → (𝜓 → 𝜑)))
13	12	com23 72	. . . . 5 ⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝜓 → (𝑥 = 𝐴 → 𝜑)))
14	13	alimi 1344	. . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → ∀𝑥(𝜓 → (𝑥 = 𝐴 → 𝜑)))
15	14	3ad2ant2 926	. . 3 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → ∀𝑥(𝜓 → (𝑥 = 𝐴 → 𝜑)))
16		19.21t 1474	. . . 4 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜓 → (𝑥 = 𝐴 → 𝜑)) ↔ (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑))))
17	16	3ad2ant1 925	. . 3 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝜓 → (𝑥 = 𝐴 → 𝜑)) ↔ (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑))))
18	15, 17	mpbid 135	. 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑)))
19	10, 18	impbid 120	1 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 98   ∧ w3a 885  ∀wal 1241   = wceq 1243  Ⅎwnf 1349  ∃wex 1381   ∈ wcel 1393

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559

This theorem is referenced by:  ceqsralt  2581