Theorem cbvex 1809

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 1572	. 2 ⊢ (𝜑 → ∀𝑦𝜑)
3		cbvex.2	. . 3 ⊢ Ⅎ𝑥𝜓
4	3	nfri 1572	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
5		cbvex.3	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
6	2, 4, 5	cbvexh 1808	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 105  Ⅎwnf 1513  ∃wex 1545

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587

This theorem depends on definitions:  df-bi 117  df-nf 1514

This theorem is referenced by:  sb8e  1910  cbvex2  1978  cbvmo  2126  mo23  2128  clelab  2366  cbvrexf  2778  issetf  2829  eqvincf  2951  rexab2  2992  cbvrexcsf  3211  abn0m  3547  rabn0m  3549  euabsn  3780  eluniab  3945  cbvopab1  4202  cbvopab2  4203  cbvopab1s  4204  intexabim  4286  iinexgm  4288  opeliunxp  4828  dfdmf  4972  dfrnf  5021  elrnmpt1  5031  cbvoprab1  6154  cbvoprab2  6155  opabex3d  6344  opabex3  6345  seq3f1olemp  10935  fsum2dlemstep  12184  bdsepnfALT  16898  strcollnfALT  16995