Theorem cbvrexv2 3139

Description: Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
cbvralv2.1	⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))
cbvralv2.2	⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)

Assertion

Ref	Expression
cbvrexv2	⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒)

Distinct variable groups:   𝑦,𝐴   𝜓,𝑦   𝑥,𝐵   𝜒,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem cbvrexv2

Step	Hyp	Ref	Expression
1		nfcv 2332	. 2 ⊢ Ⅎ𝑦𝐴
2		nfcv 2332	. 2 ⊢ Ⅎ𝑥𝐵
3		nfv 1539	. 2 ⊢ Ⅎ𝑦𝜓
4		nfv 1539	. 2 ⊢ Ⅎ𝑥𝜒
5		cbvralv2.2	. 2 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
6		cbvralv2.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))
7	1, 2, 3, 4, 5, 6	cbvrexcsf 3135	1 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1364  ∃wrex 2469

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-sbc 2978  df-csb 3073

This theorem is referenced by: (None)