Theorem sbcim1OLD

Description: Obsolete version of sbcim1 as of 26-Oct-2024. (Contributed by NM, 17-Aug-2018) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	sbcim1OLD	⊢ ( [ 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 ) )