Theorem sblim

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)

	Ref	Expression
Hypothesis	sblim.1	⊢ Ⅎ 𝑥 𝜓
Assertion	sblim	⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ ( [ 𝑦 / 𝑥 ] 𝜑 → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		sblim.1	⊢ Ⅎ 𝑥 𝜓
2		sbim	⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ ( [ 𝑦 / 𝑥 ] 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) )
3	1	sbf	⊢ ( [ 𝑦 / 𝑥 ] 𝜓 ↔ 𝜓 )
4	3	imbi2i	⊢ ( ( [ 𝑦 / 𝑥 ] 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) ↔ ( [ 𝑦 / 𝑥 ] 𝜑 → 𝜓 ) )
5	2 4	bitri	⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ ( [ 𝑦 / 𝑥 ] 𝜑 → 𝜓 ) )