Theorem cbvexdva 1976

Description: Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)

Hypothesis

Ref	Expression
cbvaldva.1	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
cbvexdva	⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))

Distinct variable groups:   𝜓,𝑦   𝜒,𝑥   𝜑,𝑥   𝜑,𝑦

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

Proof of Theorem cbvexdva

Step	Hyp	Ref	Expression
1		nfv 1574	. 2 ⊢ Ⅎ𝑦𝜑
2		nfvd 1575	. 2 ⊢ (𝜑 → Ⅎ𝑦𝜓)
3		cbvaldva.1	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
4	3	ex 115	. 2 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
5	1, 2, 4	cbvexd 1974	1 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∃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:  cbvexdvaw  1978  cbvrexdva2  2773  acexmid  5999  tfrlemi1  6476  ltexpri  7796  recexpr  7821