Theorem 2sbbid 2242

Description: Deduction doubly substituting both sides of a biconditional. (Contributed by AV, 30-Jul-2023.)

Hypotheses

Ref	Expression
sbbid.1	⊢ Ⅎ𝑥𝜑
sbbid.2	⊢ (𝜑 → (𝜓 ↔ 𝜒))
2sbbid.1	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
2sbbid	⊢ (𝜑 → ([𝑡 / 𝑥][𝑢 / 𝑦]𝜓 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜒))

Proof of Theorem 2sbbid

Step	Hyp	Ref	Expression
1		sbbid.1	. 2 ⊢ Ⅎ𝑥𝜑
2		2sbbid.1	. . 3 ⊢ Ⅎ𝑦𝜑
3		sbbid.2	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
4	2, 3	sbbid 2241	. 2 ⊢ (𝜑 → ([𝑢 / 𝑦]𝜓 ↔ [𝑢 / 𝑦]𝜒))
5	1, 4	sbbid 2241	1 ⊢ (𝜑 → ([𝑡 / 𝑥][𝑢 / 𝑦]𝜓 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜒))

Syntax hints:   → wi 4   ↔ wb 205  Ⅎwnf 1787  [wsb 2068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-12 2173

This theorem depends on definitions:  df-bi 206  df-ex 1784  df-nf 1788  df-sb 2069

This theorem is referenced by: (None)