Theorem ceqsralt 3453

Description: Restricted quantifier version of ceqsalt 3452. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

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

Proof of Theorem ceqsralt

Step	Hyp	Ref	Expression
1		df-ral 3068	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐵 → (𝑥 = 𝐴 → 𝜑)))
2		eleq1 2826	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
3	2	pm5.32ri 575	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ (𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴))
4	3	imbi1i 349	. . . . . 6 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴) → 𝜑) ↔ ((𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) → 𝜑))
5		impexp 450	. . . . . 6 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴) → 𝜑) ↔ (𝑥 ∈ 𝐵 → (𝑥 = 𝐴 → 𝜑)))
6		impexp 450	. . . . . 6 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) → 𝜑) ↔ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝜑)))
7	4, 5, 6	3bitr3i 300	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 → (𝑥 = 𝐴 → 𝜑)) ↔ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝜑)))
8	7	albii 1823	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐵 → (𝑥 = 𝐴 → 𝜑)) ↔ ∀𝑥(𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝜑)))
9		19.21v 1943	. . . 4 ⊢ (∀𝑥(𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝜑)) ↔ (𝐴 ∈ 𝐵 → ∀𝑥(𝑥 = 𝐴 → 𝜑)))
10	1, 8, 9	3bitri 296	. . 3 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ (𝐴 ∈ 𝐵 → ∀𝑥(𝑥 = 𝐴 → 𝜑)))
11	10	a1i 11	. 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ (𝐴 ∈ 𝐵 → ∀𝑥(𝑥 = 𝐴 → 𝜑))))
12		biimt 360	. . 3 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ (𝐴 ∈ 𝐵 → ∀𝑥(𝑥 = 𝐴 → 𝜑))))
13	12	3ad2ant3 1133	. 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ (𝐴 ∈ 𝐵 → ∀𝑥(𝑥 = 𝐴 → 𝜑))))
14		ceqsalt 3452	. 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
15	11, 13, 14	3bitr2d 306	1 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085  ∀wal 1537   = wceq 1539  Ⅎwnf 1787   ∈ wcel 2108  ∀wral 3063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1087  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068

This theorem is referenced by:  ceqsralvOLD  3460  cdleme32fva  38378