Theorem cbvexdva 1864

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 1476	. 2 ⊢ Ⅎ𝑦𝜑
2		nfvd 1477	. 2 ⊢ (𝜑 → Ⅎ𝑦𝜓)
3		cbvaldva.1	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
4	3	ex 114	. 2 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
5	1, 2, 4	cbvexd 1862	1 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∃wex 1436

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-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482

This theorem depends on definitions:  df-bi 116  df-nf 1405

This theorem is referenced by:  cbvrexdva2  2617  acexmid  5705  tfrlemi1  6159  ltexpri  7322  recexpr  7347