Theorem sbiev 2285

Description: Conversion of implicit substitution to explicit substitution. Version of sbie 2483 with a disjoint variable condition, not requiring ax-13 2333. (Contributed by Wolf Lammen, 18-Jan-2023.)

Hypotheses

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

Assertion

Ref	Expression
sbiev	⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem sbiev

Step	Hyp	Ref	Expression
1		equsb1v 2246	. . 3 ⊢ [𝑦 / 𝑥]𝑥 = 𝑦
2		sbiev.2	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	2	sbimi 2017	. . 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	sbfv 2245	. . 3 ⊢ ([𝑦 / 𝑥]𝜓 ↔ 𝜓)
7	6	sblbisv 2284	. 2 ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ 𝜓))
8	4, 7	mpbi 222	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 198  Ⅎwnf 1827  [wsb 2011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-10 2134  ax-12 2162

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-ex 1824  df-nf 1828  df-sb 2012

This theorem is referenced by:  sbco2vv  2289  equsb3v  2290  sbco2v  2306  elsb3v  2511  elsb4v  2514  mo4f  2582  eqsb3v  2884  clelsb3v  2888  cbvabv  2913  2reu8i  42136