Theorem ceqsralv2TEMP 36531

Description: Alternate elimination of a restricted universal quantifier, using implicit substitution. (Temporary: as soon as this Mathbox only PR is accepted, I'll open a PR to place this to the main. PM) (Contributed by Scott Fenton, 7-Dec-2020.)

Hypothesis

Ref	Expression
ceqsralv2TEMP.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
ceqsralv2TEMP	⊢ (∀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 ceqsralv2TEMP

Step	Hyp	Ref	Expression
1		ceqsralv2TEMP.1	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
2	1	notbid 318	. . . 4 ⊢ (𝑥 = 𝐴 → (¬ 𝜑 ↔ ¬ 𝜓))
3	2	ceqsrexv2TEMP 36530	. . 3 ⊢ (∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ ¬ 𝜑) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝜓))
4		rexanali 3101	. . 3 ⊢ (∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ ¬ 𝜑) ↔ ¬ ∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑))
5		annim 405	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝜓) ↔ ¬ (𝐴 ∈ 𝐵 → 𝜓))
6	3, 4, 5	3bitr3i 301	. 2 ⊢ (¬ ∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ ¬ (𝐴 ∈ 𝐵 → 𝜓))
7	6	con4bii 321	1 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ (𝐴 ∈ 𝐵 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1539   ∈ wcel 2104  ∀wral 3061  ∃wrex 3070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rex 3071

This theorem is referenced by:  ref5  36532