Theorem cbvex 1802

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 1565	. 2 ⊢ (𝜑 → ∀𝑦𝜑)
3		cbvex.2	. . 3 ⊢ Ⅎ𝑥𝜓
4	3	nfri 1565	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
5		cbvex.3	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
6	2, 4, 5	cbvexh 1801	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 105  Ⅎwnf 1506  ∃wex 1538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580

This theorem depends on definitions:  df-bi 117  df-nf 1507

This theorem is referenced by:  sb8e  1903  cbvex2  1969  cbvmo  2117  mo23  2119  clelab  2355  cbvrexf  2757  issetf  2807  eqvincf  2928  rexab2  2969  cbvrexcsf  3188  abn0m  3517  rabn0m  3519  euabsn  3736  eluniab  3900  cbvopab1  4157  cbvopab2  4158  cbvopab1s  4159  intexabim  4236  iinexgm  4238  opeliunxp  4774  dfdmf  4916  dfrnf  4965  elrnmpt1  4975  cbvoprab1  6082  cbvoprab2  6083  opabex3d  6272  opabex3  6273  seq3f1olemp  10745  fsum2dlemstep  11953  bdsepnfALT  16276  strcollnfALT  16373