Theorem cbvrexv2 3009

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 2235	. 2 ⊢ Ⅎ𝑦𝐴
2		nfcv 2235	. 2 ⊢ Ⅎ𝑥𝐵
3		nfv 1473	. 2 ⊢ Ⅎ𝑦𝜓
4		nfv 1473	. 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 3005	1 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒)

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1296  ∃wrex 2371

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-rex 2376  df-sbc 2855  df-csb 2948

This theorem is referenced by: (None)