Theorem sbied 1811

Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie 1814). (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.)

Hypotheses

Ref	Expression
sbied.1	⊢ Ⅎ𝑥𝜑
sbied.2	⊢ (𝜑 → Ⅎ𝑥𝜒)
sbied.3	⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))

Assertion

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

Proof of Theorem sbied

Step	Hyp	Ref	Expression
1		sbied.1	. . 3 ⊢ Ⅎ𝑥𝜑
2	1	nfri 1542	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3		sbied.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝜒)
4	3	nfrd 1543	. 2 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))
5		sbied.3	. 2 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
6	2, 4, 5	sbiedh 1810	1 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 105  Ⅎwnf 1483  [wsb 1785

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-i9 1553  ax-ial 1557

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786

This theorem is referenced by:  sbiedv  1812  dvelimdf  2044  cbvrald  15724