Theorem sblim 2296

Description: Substitution in an implication with a variable not free in the consequent affects only the antecedent. (Contributed by NM, 14-Nov-2013.) (Revised by Mario Carneiro, 4-Oct-2016.)

Hypothesis

Ref	Expression
sblim.1	⊢ Ⅎ𝑥𝜓

Assertion

Ref	Expression
sblim	⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → 𝜓))

Proof of Theorem sblim

Step	Hyp	Ref	Expression
1		sbim 2293	. 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
2		sblim.1	. . . 4 ⊢ Ⅎ𝑥𝜓
3	2	sbf 2258	. . 3 ⊢ ([𝑦 / 𝑥]𝜓 ↔ 𝜓)
4	3	imbi2i 336	. 2 ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ ([𝑦 / 𝑥]𝜑 → 𝜓))
5	1, 4	bitri 275	1 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 205  Ⅎwnf 1778  [wsb 2060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-10 2130  ax-12 2167

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-ex 1775  df-nf 1779  df-sb 2061

This theorem is referenced by:  sbmo  2606