Theorem ceqsralt 3513

Description: Restricted quantifier version of ceqsalt 3512. (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		biimt 360	. . . 4 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ (𝐴 ∈ 𝐵 → ∀𝑥(𝑥 = 𝐴 → 𝜑))))
2		df-ral 3059	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐵 → (𝑥 = 𝐴 → 𝜑)))
3		eleq1 2826	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
4	3	pm5.32ri 575	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ (𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴))
5	4	imbi1i 349	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴) → 𝜑) ↔ ((𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) → 𝜑))
6		impexp 450	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴) → 𝜑) ↔ (𝑥 ∈ 𝐵 → (𝑥 = 𝐴 → 𝜑)))
7		impexp 450	. . . . . . 7 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) → 𝜑) ↔ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝜑)))
8	5, 6, 7	3bitr3i 301	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 → (𝑥 = 𝐴 → 𝜑)) ↔ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝜑)))
9	8	albii 1815	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐵 → (𝑥 = 𝐴 → 𝜑)) ↔ ∀𝑥(𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝜑)))
10		19.21v 1936	. . . . 5 ⊢ (∀𝑥(𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝜑)) ↔ (𝐴 ∈ 𝐵 → ∀𝑥(𝑥 = 𝐴 → 𝜑)))
11	2, 9, 10	3bitrri 298	. . . 4 ⊢ ((𝐴 ∈ 𝐵 → ∀𝑥(𝑥 = 𝐴 → 𝜑)) ↔ ∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑))
12	1, 11	bitrdi 287	. . 3 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ ∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑)))
13	12	3ad2ant3 1134	. 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ ∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑)))
14		ceqsalt 3512	. 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
15	13, 14	bitr3d 281	1 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1534   = wceq 1536  Ⅎwnf 1779   ∈ wcel 2105  ∀wral 3058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-12 2174  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1539  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059

This theorem is referenced by:  cdleme32fva  40419