Theorem sbied 1781

Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie 1784). (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 1512	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3		sbied.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝜒)
4	3	nfrd 1513	. 2 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))
5		sbied.3	. 2 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
6	2, 4, 5	sbiedh 1780	1 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 104  Ⅎwnf 1453  [wsb 1755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-i9 1523  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756

This theorem is referenced by:  sbiedv  1782  dvelimdf  2009  cbvrald  13823