Theorem sbbidOLD 2248

Description: Obsolete version of sbbid 2246 as of 10-Jul-2023. Deduction substituting both sides of a biconditional. (Contributed by NM, 30-Jun-1993.) Remove dependency on ax-10 2145 and ax-13 2390. (Revised by Wolf Lammen, 24-Nov-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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

Assertion

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

Proof of Theorem sbbidOLD

Step	Hyp	Ref	Expression
1		sbbidOLD.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		sbbidOLD.2	. . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	2	biimpd 231	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
4	1, 3	sbimd 2245	. 2 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒))
5	2	biimprd 250	. . 3 ⊢ (𝜑 → (𝜒 → 𝜓))
6	1, 5	sbimd 2245	. 2 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜒 → [𝑦 / 𝑥]𝜓))
7	4, 6	impbid 214	1 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜒))

Syntax hints:   → wi 4   ↔ wb 208  Ⅎwnf 1784  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-ex 1781  df-nf 1785  df-sb 2070

This theorem is referenced by: (None)