Theorem cbvexv1 2347

Description: Rule used to change bound variables, using implicit substitution. Version of cbvex 2404 with a disjoint variable condition, which does not require ax-13 2377. See cbvexvw 2039 for a version with two disjoint variable conditions, requiring fewer axioms, and cbvexv 2406 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 1859	. . . 4 ⊢ Ⅎ𝑦 ¬ 𝜑
3		cbvalv1.nf2	. . . . 5 ⊢ Ⅎ𝑥𝜓
4	3	nfn 1859	. . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓
5		cbvalv1.1	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
6	5	notbid 318	. . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
7	2, 4, 6	cbvalv1 2346	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ∀𝑦 ¬ 𝜓)
8		alnex 1783	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
9		alnex 1783	. . 3 ⊢ (∀𝑦 ¬ 𝜓 ↔ ¬ ∃𝑦𝜓)
10	7, 8, 9	3bitr3i 301	. 2 ⊢ (¬ ∃𝑥𝜑 ↔ ¬ ∃𝑦𝜓)
11	10	con4bii 321	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206  ∀wal 1540  ∃wex 1781  Ⅎwnf 1785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-11 2163  ax-12 2185

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1782  df-nf 1786

This theorem is referenced by:  sb8ef  2360  exsb  2364  mof  2564  euf  2577  cbveuw  2607  eqvincf  3593  rexab2  3646  euabsn  4671  eluniab  4865  cbvopab1  5160  cbvopab1g  5161  cbvopab2  5162  cbvopab1s  5163  axrep1  5214  axrep2  5216  axrep4OLD  5220  opeliunxp  5698  opeliun2xp  5699  dfdmf  5852  dfrnf  5906  elrnmpt1  5916  cbvoprab1  7454  cbvoprab2  7455  opabex3d  7918  opabex3rd  7919  opabex3  7920  zfcndrep  10537  fsum2dlem  15732  fprod2dlem  15945  2ndresdju  32722  bnj1146  34933  bnj607  35058  bnj1228  35153  fineqvrep  35258  poimirlem26  37967  sbcexf  38436  elunif  45447  stoweidlem46  46474