Theorem cbvrexcsf 3895

Description: A more general version of cbvrexf 3347 that has no distinct variable restrictions. Changes bound variables using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2402. (Contributed by Andrew Salmon, 13-Jul-2011.) (Proof shortened by Mario Carneiro, 7-Dec-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cbvralcsf.1	⊢ Ⅎ𝑦𝐴
cbvralcsf.2	⊢ Ⅎ𝑥𝐵
cbvralcsf.3	⊢ Ⅎ𝑦𝜑
cbvralcsf.4	⊢ Ⅎ𝑥𝜓
cbvralcsf.5	⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
cbvralcsf.6	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvrexcsf	⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜓)

Proof of Theorem cbvrexcsf

Step	Hyp	Ref	Expression
1		cbvralcsf.1	. . . 4 ⊢ Ⅎ𝑦𝐴
2		cbvralcsf.2	. . . 4 ⊢ Ⅎ𝑥𝐵
3		cbvralcsf.3	. . . . 5 ⊢ Ⅎ𝑦𝜑
4	3	nfn 1876	. . . 4 ⊢ Ⅎ𝑦 ¬ 𝜑
5		cbvralcsf.4	. . . . 5 ⊢ Ⅎ𝑥𝜓
6	5	nfn 1876	. . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓
7		cbvralcsf.5	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
8		cbvralcsf.6	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
9	8	notbid 320	. . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
10	1, 2, 4, 6, 7, 9	cbvralcsf 3894	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐵 ¬ 𝜓)
11	10	notbii 322	. 2 ⊢ (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜓)
12		dfrex2 3088	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑)
13		dfrex2 3088	. 2 ⊢ (∃𝑦 ∈ 𝐵 𝜓 ↔ ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜓)
14	11, 12, 13	3bitr4i 305	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   = wceq 1559  Ⅎwnf 1802  Ⅎwnfc 2908  ∀wral 3075  ∃wrex 3085

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-13 2402  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3076  df-rex 3086  df-sbc 3745  df-csb 3853

This theorem is referenced by:  cbvrexv2  3899