Theorem sbiedv 1730

Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie 1732). (Contributed by NM, 7-Jan-2017.)

Hypothesis

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

Assertion

Ref	Expression
sbiedv	⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒))

Distinct variable groups:   𝜑,𝑥   𝜒,𝑥

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

Proof of Theorem sbiedv

Step	Hyp	Ref	Expression
1		nfv 1476	. 2 ⊢ Ⅎ𝑥𝜑
2		nfvd 1477	. 2 ⊢ (𝜑 → Ⅎ𝑥𝜒)
3		sbiedv.1	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
4	3	ex 114	. 2 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
5	1, 2, 4	sbied 1729	1 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  [wsb 1703

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-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482

This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704

This theorem is referenced by:  acexmid  5705