Theorem sbievOLD 2296

Description: Obsolete proof of sbiev 2295 as of 18-Jul-2023. (Contributed by Wolf Lammen, 18-Jan-2023.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
sbiev.1	⊢ Ⅎ𝑥𝜓
sbiev.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
sbievOLD	⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem sbievOLD

Step	Hyp	Ref	Expression
1		equsb1v 2079	. . 3 ⊢ [𝑦 / 𝑥]𝑥 = 𝑦
2		sbiev.2	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	2	sbimi 2052	. . 3 ⊢ ([𝑦 / 𝑥]𝑥 = 𝑦 → [𝑦 / 𝑥](𝜑 ↔ 𝜓))
4	1, 3	ax-mp 5	. 2 ⊢ [𝑦 / 𝑥](𝜑 ↔ 𝜓)
5		sbiev.1	. . . 4 ⊢ Ⅎ𝑥𝜓
6	5	sbf 2234	. . 3 ⊢ ([𝑦 / 𝑥]𝜓 ↔ 𝜓)
7	6	sblbisvOLD 2294	. 2 ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ 𝜓))
8	4, 7	mpbi 231	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 207  Ⅎwnf 1765  [wsb 2042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-10 2112  ax-12 2141

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-ex 1762  df-nf 1766  df-sb 2043

This theorem is referenced by: (None)