Theorem cbvrexf 3142

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 1768	. . . 4 ⊢ Ⅎ𝑦 ¬ 𝜑
5		cbvralf.4	. . . . 5 ⊢ Ⅎ𝑥𝜓
6	5	nfn 1768	. . . 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 3141	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝜓)
10	9	notbii 309	. 2 ⊢ (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ 𝜓)
11		dfrex2 2979	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑)
12		dfrex2 2979	. 2 ⊢ (∃𝑦 ∈ 𝐴 𝜓 ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ 𝜓)
13	10, 11, 12	3bitr4i 291	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195  Ⅎwnf 1699  Ⅎwnfc 2738  ∀wral 2896  ∃wrex 2897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902

This theorem is referenced by:  cbvrex  3144  reusv2lem4  4798  reusv2  4800  nnwof  11630  cbviunf  28755  ac6sf2  28810  dfimafnf  28817  aciunf1lem  28844  bnj1400  30160  phpreu  32563  poimirlem26  32605  indexa  32698  evth2f  38197  fvelrnbf  38200  evthf  38209  eliin2f  38316  stoweidlem34  38927  ovnlerp  39452