Theorem sblim 2017

Description: Substitution with a variable not free in 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 2013	. 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
2		sblim.1	. . . 4 ⊢ Ⅎ𝑥𝜓
3	2	sbf 1830	. . 3 ⊢ ([𝑦 / 𝑥]𝜓 ↔ 𝜓)
4	3	imbi2i 226	. 2 ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ ([𝑦 / 𝑥]𝜑 → 𝜓))
5	1, 4	bitri 184	1 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 105  Ⅎwnf 1513  [wsb 1815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816

This theorem is referenced by:  sbnf2  2041  sbmo  2146