Theorem cbvex 1749

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
cbvex.1	⊢ Ⅎ𝑦𝜑
cbvex.2	⊢ Ⅎ𝑥𝜓
cbvex.3	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvex	⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Proof of Theorem cbvex

Step	Hyp	Ref	Expression
1		cbvex.1	. . 3 ⊢ Ⅎ𝑦𝜑
2	1	nfri 1512	. 2 ⊢ (𝜑 → ∀𝑦𝜑)
3		cbvex.2	. . 3 ⊢ Ⅎ𝑥𝜓
4	3	nfri 1512	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
5		cbvex.3	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
6	2, 4, 5	cbvexh 1748	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 104  Ⅎwnf 1453  ∃wex 1485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-nf 1454

This theorem is referenced by:  sb8e  1850  cbvex2  1915  cbvmo  2059  mo23  2060  clelab  2296  cbvrexf  2690  issetf  2737  eqvincf  2855  rexab2  2896  cbvrexcsf  3112  abn0m  3440  rabn0m  3442  euabsn  3653  eluniab  3808  cbvopab1  4062  cbvopab2  4063  cbvopab1s  4064  intexabim  4138  iinexgm  4140  opeliunxp  4666  dfdmf  4804  dfrnf  4852  elrnmpt1  4862  cbvoprab1  5925  cbvoprab2  5926  opabex3d  6100  opabex3  6101  seq3f1olemp  10458  fsum2dlemstep  11397  bdsepnfALT  13924  strcollnfALT  14021