Theorem cbvrexf 2831

Description: Rule used to change bound variables, using implicit substitution. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 9-Oct-2016.)

Hypotheses

Ref	Expression
cbvralf.1	⊢ ℲxA
cbvralf.2	⊢ ℲyA
cbvralf.3	⊢ Ⅎyφ
cbvralf.4	⊢ Ⅎxψ
cbvralf.5	⊢ (x = y → (φ ↔ ψ))

Assertion

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

Proof of Theorem cbvrexf

Step	Hyp	Ref	Expression
1		cbvralf.1	. . . 4 ⊢ ℲxA
2		cbvralf.2	. . . 4 ⊢ ℲyA
3		cbvralf.3	. . . . 5 ⊢ Ⅎyφ
4	3	nfn 1793	. . . 4 ⊢ Ⅎy ¬ φ
5		cbvralf.4	. . . . 5 ⊢ Ⅎxψ
6	5	nfn 1793	. . . 4 ⊢ Ⅎx ¬ ψ
7		cbvralf.5	. . . . 5 ⊢ (x = y → (φ ↔ ψ))
8	7	notbid 285	. . . 4 ⊢ (x = y → (¬ φ ↔ ¬ ψ))
9	1, 2, 4, 6, 8	cbvralf 2830	. . 3 ⊢ (∀x ∈ A ¬ φ ↔ ∀y ∈ A ¬ ψ)
10	9	notbii 287	. 2 ⊢ (¬ ∀x ∈ A ¬ φ ↔ ¬ ∀y ∈ A ¬ ψ)
11		dfrex2 2628	. 2 ⊢ (∃x ∈ A φ ↔ ¬ ∀x ∈ A ¬ φ)
12		dfrex2 2628	. 2 ⊢ (∃y ∈ A ψ ↔ ¬ ∀y ∈ A ¬ ψ)
13	10, 11, 12	3bitr4i 268	1 ⊢ (∃x ∈ A φ ↔ ∃y ∈ A 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-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621

This theorem is referenced by:  cbvrex  2833