Theorem ceqsalt 3462

Description: Closed theorem version of ceqsalg 3464. (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 2820	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
2	1	3ad2ant3 1134	. . 3 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → ∃𝑥 𝑥 = 𝐴)
3		biimp 214	. . . . . . 7 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
4	3	imim3i 64	. . . . . 6 ⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → ((𝑥 = 𝐴 → 𝜑) → (𝑥 = 𝐴 → 𝜓)))
5	4	al2imi 1818	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (∀𝑥(𝑥 = 𝐴 → 𝜑) → ∀𝑥(𝑥 = 𝐴 → 𝜓)))
6	5	3ad2ant2 1133	. . . 4 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) → ∀𝑥(𝑥 = 𝐴 → 𝜓)))
7		19.23t 2203	. . . . 5 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ (∃𝑥 𝑥 = 𝐴 → 𝜓)))
8	7	3ad2ant1 1132	. . . 4 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ (∃𝑥 𝑥 = 𝐴 → 𝜓)))
9	6, 8	sylibd 238	. . 3 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) → (∃𝑥 𝑥 = 𝐴 → 𝜓)))
10	2, 9	mpid 44	. 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) → 𝜓))
11		biimpr 219	. . . . . . 7 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
12	11	imim2i 16	. . . . . 6 ⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝑥 = 𝐴 → (𝜓 → 𝜑)))
13	12	com23 86	. . . . 5 ⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝜓 → (𝑥 = 𝐴 → 𝜑)))
14	13	alimi 1814	. . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → ∀𝑥(𝜓 → (𝑥 = 𝐴 → 𝜑)))
15	14	3ad2ant2 1133	. . 3 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → ∀𝑥(𝜓 → (𝑥 = 𝐴 → 𝜑)))
16		19.21t 2199	. . . 4 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜓 → (𝑥 = 𝐴 → 𝜑)) ↔ (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑))))
17	16	3ad2ant1 1132	. . 3 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝜓 → (𝑥 = 𝐴 → 𝜑)) ↔ (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑))))
18	15, 17	mpbid 231	. 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑)))
19	10, 18	impbid 211	1 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1086  ∀wal 1537   = wceq 1539  ∃wex 1782  Ⅎwnf 1786   ∈ wcel 2106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1088  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-clel 2816

This theorem is referenced by:  ceqsralt  3463  ceqsalg  3464