Theorem sbcim1

Description: Distribution of class substitution over implication. One direction of sbcimg that holds for proper classes. (Contributed by NM, 17-Aug-2018)

	Ref	Expression
Assertion	sbcim1	⊢ ( [ 𝐴 / 𝑥 ] ( 𝜑 → 𝜓 ) → ( [ 𝐴 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		sbcex	⊢ ( [ 𝐴 / 𝑥 ] ( 𝜑 → 𝜓 ) → 𝐴 ∈ V )
2		sbcimg	⊢ ( 𝐴 ∈ V → ( [ 𝐴 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ ( [ 𝐴 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] 𝜓 ) ) )
3	2	biimpd	⊢ ( 𝐴 ∈ V → ( [ 𝐴 / 𝑥 ] ( 𝜑 → 𝜓 ) → ( [ 𝐴 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] 𝜓 ) ) )
4	1 3	mpcom	⊢ ( [ 𝐴 / 𝑥 ] ( 𝜑 → 𝜓 ) → ( [ 𝐴 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] 𝜓 ) )