Theorem ceqsralt 2766

Description: Restricted quantifier version of ceqsalt 2765. (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 2460	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐵 → (𝑥 = 𝐴 → 𝜑)))
2		eleq1 2240	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
3	2	pm5.32ri 455	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ (𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴))
4	3	imbi1i 238	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴) → 𝜑) ↔ ((𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) → 𝜑))
5		impexp 263	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴) → 𝜑) ↔ (𝑥 ∈ 𝐵 → (𝑥 = 𝐴 → 𝜑)))
6		impexp 263	. . . . . . 7 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) → 𝜑) ↔ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝜑)))
7	4, 5, 6	3bitr3i 210	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 → (𝑥 = 𝐴 → 𝜑)) ↔ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝜑)))
8	7	albii 1470	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐵 → (𝑥 = 𝐴 → 𝜑)) ↔ ∀𝑥(𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝜑)))
9	8	a1i 9	. . . 4 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → (∀𝑥(𝑥 ∈ 𝐵 → (𝑥 = 𝐴 → 𝜑)) ↔ ∀𝑥(𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝜑))))
10	1, 9	bitrid 192	. . 3 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ ∀𝑥(𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝜑))))
11		19.21v 1873	. . 3 ⊢ (∀𝑥(𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝜑)) ↔ (𝐴 ∈ 𝐵 → ∀𝑥(𝑥 = 𝐴 → 𝜑)))
12	10, 11	bitrdi 196	. 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ (𝐴 ∈ 𝐵 → ∀𝑥(𝑥 = 𝐴 → 𝜑))))
13		biimt 241	. . 3 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ (𝐴 ∈ 𝐵 → ∀𝑥(𝑥 = 𝐴 → 𝜑))))
14	13	3ad2ant3 1020	. 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ (𝐴 ∈ 𝐵 → ∀𝑥(𝑥 = 𝐴 → 𝜑))))
15		ceqsalt 2765	. 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
16	12, 14, 15	3bitr2d 216	1 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978  ∀wal 1351   = wceq 1353  Ⅎwnf 1460   ∈ wcel 2148  ∀wral 2455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-ral 2460  df-v 2741

This theorem is referenced by:  ceqsralv  2770