Theorem ceqsralv 3503

Description: Restricted quantifier version of ceqsalv 3502. (Contributed by NM, 21-Jun-2013.) Avoid ax-9 2159, ax-12 2219, ax-ext 2741. (Revised by SN, 8-Sep-2024.)

Hypothesis

Ref	Expression
ceqsralv.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
ceqsralv	⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ 𝜓))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ceqsralv

Step	Hyp	Ref	Expression
1		ceqsralv.2	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
2	1	pm5.74i 274	. . 3 ⊢ ((𝑥 = 𝐴 → 𝜑) ↔ (𝑥 = 𝐴 → 𝜓))
3	2	ralbii 3117	. 2 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ ∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜓))
4		r19.23v 3198	. . 3 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜓) ↔ (∃𝑥 ∈ 𝐵 𝑥 = 𝐴 → 𝜓))
5		risset 3246	. . . 4 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝑥 = 𝐴)
6		pm5.5 364	. . . 4 ⊢ (∃𝑥 ∈ 𝐵 𝑥 = 𝐴 → ((∃𝑥 ∈ 𝐵 𝑥 = 𝐴 → 𝜓) ↔ 𝜓))
7	5, 6	sylbi 220	. . 3 ⊢ (𝐴 ∈ 𝐵 → ((∃𝑥 ∈ 𝐵 𝑥 = 𝐴 → 𝜓) ↔ 𝜓))
8	4, 7	bitrid 286	. 2 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜓) ↔ 𝜓))
9	3, 8	bitrid 286	1 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-clel 2844  df-ral 3086  df-rex 3096

This theorem is referenced by:  eqreu  3701  sqrt2irr  16305  acsfn  17715  ovolgelb  25608  fsuppind  43214