Theorem cbvexv1 2376

Description: Rule used to change bound variables, using implicit substitution. Version of cbvex 2433 with a disjoint variable condition, which does not require ax-13 2406. See cbvexvw 2060 for a version with two disjoint variable conditions, requiring fewer axioms, and cbvexv 2435 for another variant. (Contributed by NM, 21-Jun-1993.) (Revised by BJ, 31-May-2019.)

Hypotheses

Ref	Expression
cbvalv1.nf1	⊢ Ⅎ𝑦𝜑
cbvalv1.nf2	⊢ Ⅎ𝑥𝜓
cbvalv1.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvexv1	⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbvexv1

Step	Hyp	Ref	Expression
1		cbvalv1.nf1	. . . . 5 ⊢ Ⅎ𝑦𝜑
2	1	nfn 1880	. . . 4 ⊢ Ⅎ𝑦 ¬ 𝜑
3		cbvalv1.nf2	. . . . 5 ⊢ Ⅎ𝑥𝜓
4	3	nfn 1880	. . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓
5		cbvalv1.1	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
6	5	notbid 321	. . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
7	2, 4, 6	cbvalv1 2375	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ∀𝑦 ¬ 𝜓)
8		alnex 1804	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
9		alnex 1804	. . 3 ⊢ (∀𝑦 ¬ 𝜓 ↔ ¬ ∃𝑦𝜓)
10	7, 8, 9	3bitr3i 304	. 2 ⊢ (¬ ∃𝑥𝜑 ↔ ¬ ∃𝑦𝜓)
11	10	con4bii 324	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1561  ∃wex 1802  Ⅎwnf 1806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-11 2194  ax-12 2215

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1803  df-nf 1807

This theorem is referenced by:  sb8ef  2389  exsb  2393  mof  2593  euf  2606  cbveuw  2636  eqvincf  3612  rexab2  3665  euabsn  4688  eluniab  4882  cbvopab1  5179  cbvopab1g  5180  cbvopab2  5181  cbvopab1s  5182  axrep1  5233  axrep2  5235  axrep4OLD  5239  opeliunxp  5719  opeliun2xp  5720  dfdmf  5877  dfrnf  5931  elrnmpt1  5941  cbvoprab1  7487  cbvoprab2  7488  opabex3d  7950  opabex3rd  7951  opabex3  7952  zfcndrep  10587  fsum2dlem  15811  fprod2dlem  16024  2ndresdju  32906  bnj1146  35096  bnj607  35221  bnj1228  35316  fineqvrep  35422  poimirlem26  38157  sbcexf  38626  elunif  45594  stoweidlem46  46618