Theorem cbvrexf 3347

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2402. Use the weaker cbvrexfw 3302 when possible. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 9-Oct-2016.) (New usage is discouraged.)

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 1876	. . . 4 ⊢ Ⅎ𝑦 ¬ 𝜑
5		cbvralf.4	. . . . 5 ⊢ Ⅎ𝑥𝜓
6	5	nfn 1876	. . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓
7		cbvralf.5	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
8	7	notbid 320	. . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
9	1, 2, 4, 6, 8	cbvralf 3346	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝜓)
10	9	notbii 322	. 2 ⊢ (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ 𝜓)
11		dfrex2 3088	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑)
12		dfrex2 3088	. 2 ⊢ (∃𝑦 ∈ 𝐴 𝜓 ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ 𝜓)
13	10, 11, 12	3bitr4i 305	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208  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-10 2174  ax-11 2190  ax-12 2211  ax-13 2402

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-clel 2836  df-nfc 2910  df-ral 3076  df-rex 3086

This theorem is referenced by:  cbvrex  3349