Theorem sbcani

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

	Ref	Expression
Hypotheses	sbcani.1	⊢ ( [ 𝐴 / 𝑥 ] 𝜑 ↔ 𝜒 )
	sbcani.2	⊢ ( [ 𝐴 / 𝑥 ] 𝜓 ↔ 𝜂 )
Assertion	sbcani	⊢ ( [ 𝐴 / 𝑥 ] ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜒 ∧ 𝜂 ) )

Proof

Step	Hyp	Ref	Expression
1		sbcani.1	⊢ ( [ 𝐴 / 𝑥 ] 𝜑 ↔ 𝜒 )
2		sbcani.2	⊢ ( [ 𝐴 / 𝑥 ] 𝜓 ↔ 𝜂 )
3		sbcan	⊢ ( [ 𝐴 / 𝑥 ] ( 𝜑 ∧ 𝜓 ) ↔ ( [ 𝐴 / 𝑥 ] 𝜑 ∧ [ 𝐴 / 𝑥 ] 𝜓 ) )
4	1 2	anbi12i	⊢ ( ( [ 𝐴 / 𝑥 ] 𝜑 ∧ [ 𝐴 / 𝑥 ] 𝜓 ) ↔ ( 𝜒 ∧ 𝜂 ) )
5	3 4	bitri	⊢ ( [ 𝐴 / 𝑥 ] ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜒 ∧ 𝜂 ) )