Theorem ceqsralv 3520

Description: Restricted quantifier version of ceqsalv 3519. (Contributed by NM, 21-Jun-2013.) Avoid ax-9 2116, ax-12 2175, ax-ext 2706. (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 271	. . 3 ⊢ ((𝑥 = 𝐴 → 𝜑) ↔ (𝑥 = 𝐴 → 𝜓))
3	2	ralbii 3091	. 2 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ ∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜓))
4		r19.23v 3181	. . 3 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜓) ↔ (∃𝑥 ∈ 𝐵 𝑥 = 𝐴 → 𝜓))
5		risset 3231	. . . 4 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝑥 = 𝐴)
6		pm5.5 361	. . . 4 ⊢ (∃𝑥 ∈ 𝐵 𝑥 = 𝐴 → ((∃𝑥 ∈ 𝐵 𝑥 = 𝐴 → 𝜓) ↔ 𝜓))
7	5, 6	sylbi 217	. . 3 ⊢ (𝐴 ∈ 𝐵 → ((∃𝑥 ∈ 𝐵 𝑥 = 𝐴 → 𝜓) ↔ 𝜓))
8	4, 7	bitrid 283	. 2 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜓) ↔ 𝜓))
9	3, 8	bitrid 283	1 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-clel 2814  df-ral 3060  df-rex 3069

This theorem is referenced by:  eqreu  3738  sqrt2irr  16282  acsfn  17704  ovolgelb  25529  fsuppind  42577