Theorem sbcimi

Description: Distribution of class substitution over implication, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019)

	Ref	Expression
Hypotheses	sbcimi.1	⊢ 𝐴 ∈ V
	sbcimi.2	⊢ ( [ 𝐴 / 𝑥 ] 𝜑 ↔ 𝜒 )
	sbcimi.3	⊢ ( [ 𝐴 / 𝑥 ] 𝜓 ↔ 𝜂 )
Assertion	sbcimi	⊢ ( [ 𝐴 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ ( 𝜒 → 𝜂 ) )

Proof

Step	Hyp	Ref	Expression
1		sbcimi.1	⊢ 𝐴 ∈ V
2		sbcimi.2	⊢ ( [ 𝐴 / 𝑥 ] 𝜑 ↔ 𝜒 )
3		sbcimi.3	⊢ ( [ 𝐴 / 𝑥 ] 𝜓 ↔ 𝜂 )
4		sbcimg	⊢ ( 𝐴 ∈ V → ( [ 𝐴 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ ( [ 𝐴 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] 𝜓 ) ) )
5	1 4	ax-mp	⊢ ( [ 𝐴 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ ( [ 𝐴 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] 𝜓 ) )
6	2 3	imbi12i	⊢ ( ( [ 𝐴 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] 𝜓 ) ↔ ( 𝜒 → 𝜂 ) )
7	5 6	bitri	⊢ ( [ 𝐴 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ ( 𝜒 → 𝜂 ) )