Theorem wl-sbrimt 37507

Description: Substitution with a variable not free in antecedent affects only the consequent. Closed form of sbrim 2303. (Contributed by Wolf Lammen, 26-Jul-2019.)

Assertion

Ref	Expression
wl-sbrimt	⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)))

Proof of Theorem wl-sbrimt

Step	Hyp	Ref	Expression
1		sbim 2302	. 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
2		sbft 2269	. . 3 ⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑))
3	2	imbi1d 341	. 2 ⊢ (Ⅎ𝑥𝜑 → (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)))
4	1, 3	bitrid 283	1 ⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)))

Syntax hints:   → wi 4   ↔ wb 206  Ⅎwnf 1782  [wsb 2063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-10 2140  ax-12 2176

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-nf 1783  df-sb 2064

This theorem is referenced by: (None)