Theorem sbievOLD 2319

Description: Obsolete version of sbiev 2318 as of 24-Aug-2025. (Contributed by NM, 30-Jun-1994.) (Revised by Wolf Lammen, 18-Jan-2023.) Remove dependence on ax-10 2141 and shorten proof. (Revised by BJ, 18-Jul-2023.) (Proof modification is discouraged.) (New usage 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		sb6 2085	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
2		sbiev.1	. . 3 ⊢ Ⅎ𝑥𝜓
3		sbiev.2	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	2, 3	equsalv 2268	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)
5	1, 4	bitri 275	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535  Ⅎwnf 1781  [wsb 2064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-nf 1782  df-sb 2065

This theorem is referenced by: (None)