Theorem sblim 2307

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 2304	. 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
2		sblim.1	. . . 4 ⊢ Ⅎ𝑥𝜓
3	2	sbf 2267	. . 3 ⊢ ([𝑦 / 𝑥]𝜓 ↔ 𝜓)
4	3	imbi2i 339	. 2 ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ ([𝑦 / 𝑥]𝜑 → 𝜓))
5	1, 4	bitri 278	1 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 209  Ⅎwnf 1791  [wsb 2070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-10 2141  ax-12 2175

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-ex 1788  df-nf 1792  df-sb 2071

This theorem is referenced by:  sbmo  2615