Theorem cbvrexcsf 3200

Description: A more general version of cbvrexf 2831 that has no distinct variable restrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.) (Proof shortened by Mario Carneiro, 7-Dec-2014.)

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 1793	. . . 4 ⊢ Ⅎy ¬ φ
5		cbvralcsf.4	. . . . 5 ⊢ Ⅎxψ
6	5	nfn 1793	. . . 4 ⊢ Ⅎx ¬ ψ
7		cbvralcsf.5	. . . 4 ⊢ (x = y → A = B)
8		cbvralcsf.6	. . . . 5 ⊢ (x = y → (φ ↔ ψ))
9	8	notbid 285	. . . 4 ⊢ (x = y → (¬ φ ↔ ¬ ψ))
10	1, 2, 4, 6, 7, 9	cbvralcsf 3199	. . 3 ⊢ (∀x ∈ A ¬ φ ↔ ∀y ∈ B ¬ ψ)
11	10	notbii 287	. 2 ⊢ (¬ ∀x ∈ A ¬ φ ↔ ¬ ∀y ∈ B ¬ ψ)
12		dfrex2 2628	. 2 ⊢ (∃x ∈ A φ ↔ ¬ ∀x ∈ A ¬ φ)
13		dfrex2 2628	. 2 ⊢ (∃y ∈ B ψ ↔ ¬ ∀y ∈ B ¬ ψ)
14	11, 12, 13	3bitr4i 268	1 ⊢ (∃x ∈ A φ ↔ ∃y ∈ B ψ)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  Ⅎwnf 1544   = wceq 1642  Ⅎwnfc 2477  ∀wral 2615  ∃wrex 2616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-sbc 3048  df-csb 3138

This theorem is referenced by:  cbvrexv2  3204