Theorem wl-sblimt 36113

Description: Substitution with a variable not free in antecedent affects only the consequent. Closed form of sbrim 2300. (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 2299	. 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
2		sbft 2261	. . 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 282	1 ⊢ (Ⅎ𝑥𝜓 → ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 205  Ⅎwnf 1785  [wsb 2067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ex 1782  df-nf 1786  df-sb 2068

This theorem is referenced by: (None)