Theorem sbbidh 1833

Description: Deduction substituting both sides of a biconditional. New proofs should use sbbid 1834 instead. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Hypotheses

Ref	Expression
sbbidh.1	⊢ (𝜑 → ∀𝑥𝜑)
sbbidh.2	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

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

Proof of Theorem sbbidh

Step	Hyp	Ref	Expression
1		sbbidh.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
2		sbbidh.2	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	1, 2	alrimih 1457	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒))
4		spsbbi 1832	. 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜒))
5	3, 4	syl 14	1 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜒))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1341  [wsb 1750

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 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-sb 1751

This theorem is referenced by:  sbcomxyyz  1960