Theorem cbvrexf 3196

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 1824	. . . 4 ⊢ Ⅎ𝑦 ¬ 𝜑
5		cbvralf.4	. . . . 5 ⊢ Ⅎ𝑥𝜓
6	5	nfn 1824	. . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓
7		cbvralf.5	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
8	7	notbid 307	. . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
9	1, 2, 4, 6, 8	cbvralf 3195	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝜓)
10	9	notbii 309	. 2 ⊢ (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ 𝜓)
11		dfrex2 3025	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑)
12		dfrex2 3025	. 2 ⊢ (∃𝑦 ∈ 𝐴 𝜓 ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ 𝜓)
13	10, 11, 12	3bitr4i 292	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196  Ⅎwnf 1748  Ⅎwnfc 2780  ∀wral 2941  ∃wrex 2942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947

This theorem is referenced by:  cbvrex  3198  reusv2lem4  4902  reusv2  4904  nnwof  11792  cbviunf  29498  ac6sf2  29557  dfimafnf  29564  aciunf1lem  29590  bnj1400  31032  phpreu  33523  poimirlem26  33565  indexa  33658  evth2f  39488  fvelrnbf  39491  evthf  39500  eliin2f  39601  stoweidlem34  40569  ovnlerp  41097