Theorem cbvrex 2736

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Hypotheses

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

Assertion

Ref	Expression
cbvrex	⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)

Distinct variable groups:   𝑥,𝐴   𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbvrex

Step	Hyp	Ref	Expression
1		nfcv 2349	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2349	. 2 ⊢ Ⅎ𝑦𝐴
3		cbvral.1	. 2 ⊢ Ⅎ𝑦𝜑
4		cbvral.2	. 2 ⊢ Ⅎ𝑥𝜓
5		cbvral.3	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
6	1, 2, 3, 4, 5	cbvrexf 2732	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 105  Ⅎwnf 1484  ∃wrex 2486

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-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rex 2491

This theorem is referenced by:  cbvrmo  2738  cbvrexv  2740  cbvrexsv  2756  cbviun  3966  disjiun  4042  rexxpf  4829  isarep1  5365  rexrnmpt  5730  elabrex  5833