Theorem ceqsralbv 3618

Description: Elimination of a restricted universal quantifier, using implicit substitution. (Contributed by Scott Fenton, 7-Dec-2020.)

Hypothesis

Ref	Expression
ceqsrexv.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

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

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

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ceqsralbv

Step	Hyp	Ref	Expression
1		ceqsrexv.1	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
2	1	notbid 320	. . . 4 ⊢ (𝑥 = 𝐴 → (¬ 𝜑 ↔ ¬ 𝜓))
3	2	ceqsrexbv 3617	. . 3 ⊢ (∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ ¬ 𝜑) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝜓))
4		rexanali 3118	. . 3 ⊢ (∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ ¬ 𝜑) ↔ ¬ ∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑))
5		annim 407	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝜓) ↔ ¬ (𝐴 ∈ 𝐵 → 𝜓))
6	3, 4, 5	3bitr3i 303	. 2 ⊢ (¬ ∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ ¬ (𝐴 ∈ 𝐵 → 𝜓))
7	6	con4bii 323	1 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ (𝐴 ∈ 𝐵 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1562   ∈ wcel 2144  ∀wral 3078  ∃wrex 3088

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1565  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rex 3089

This theorem is referenced by:  ref5  38823