Theorem sbi2

Description: Introduction of implication into substitution. (Contributed by NM, 14-May-1993)

	Ref	Expression
Assertion	sbi2	⊢ ( ( [ 𝑦 / 𝑥 ] 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) → [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		sbn	⊢ ( [ 𝑦 / 𝑥 ] ¬ 𝜑 ↔ ¬ [ 𝑦 / 𝑥 ] 𝜑 )
2		pm2.21	⊢ ( ¬ 𝜑 → ( 𝜑 → 𝜓 ) )
3	2	sbimi	⊢ ( [ 𝑦 / 𝑥 ] ¬ 𝜑 → [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) )
4	1 3	sylbir	⊢ ( ¬ [ 𝑦 / 𝑥 ] 𝜑 → [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) )
5		ax-1	⊢ ( 𝜓 → ( 𝜑 → 𝜓 ) )
6	5	sbimi	⊢ ( [ 𝑦 / 𝑥 ] 𝜓 → [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) )
7	4 6	ja	⊢ ( ( [ 𝑦 / 𝑥 ] 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) → [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) )