Theorem cbvexv1 2340

Description: Rule used to change bound variables, using implicit substitution. Version of cbvex 2397 with a disjoint variable condition, which does not require ax-13 2370. See cbvexvw 2037 for a version with two disjoint variable conditions, requiring fewer axioms, and cbvexv 2399 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 1857	. . . 4 ⊢ Ⅎ𝑦 ¬ 𝜑
3		cbvalv1.nf2	. . . . 5 ⊢ Ⅎ𝑥𝜓
4	3	nfn 1857	. . . 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 2339	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ∀𝑦 ¬ 𝜓)
8		alnex 1781	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
9		alnex 1781	. . 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 1538  ∃wex 1779  Ⅎwnf 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-11 2158  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1780  df-nf 1784

This theorem is referenced by:  sb8ef  2353  exsb  2357  mof  2556  euf  2569  cbveuw  2599  eqvincf  3616  rexab2  3670  euabsn  4690  eluniab  4885  cbvopab1  5181  cbvopab1g  5182  cbvopab2  5183  cbvopab1s  5184  axrep1  5235  axrep2  5237  axrep4OLD  5241  opeliunxp  5705  opeliun2xp  5706  dfdmf  5860  dfrnf  5914  elrnmpt1  5924  cbvoprab1  7476  cbvoprab2  7477  opabex3d  7944  opabex3rd  7945  opabex3  7946  zfcndrep  10567  fsum2dlem  15736  fprod2dlem  15946  2ndresdju  32573  bnj1146  34781  bnj607  34906  bnj1228  35001  fineqvrep  35085  poimirlem26  37640  sbcexf  38109  elunif  45010  stoweidlem46  46044