Theorem wl-sblimt 37531

Description: Substitution with a variable not free in antecedent affects only the consequent. Closed form of sbrim 2304. (Contributed by Wolf Lammen, 26-Jul-2019.)

Assertion

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

Proof of Theorem wl-sblimt

Step	Hyp	Ref	Expression
1		sbim 2303	. 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
2		sbft 2270	. . 3 ⊢ (Ⅎ𝑥𝜓 → ([𝑦 / 𝑥]𝜓 ↔ 𝜓))
3	2	imbi2d 340	. 2 ⊢ (Ⅎ𝑥𝜓 → (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ ([𝑦 / 𝑥]𝜑 → 𝜓)))
4	1, 3	bitrid 283	1 ⊢ (Ⅎ𝑥𝜓 → ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 206  Ⅎwnf 1783  [wsb 2065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-10 2142  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784  df-sb 2066

This theorem is referenced by: (None)