Theorem sbc19.21g

Description: Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004)

	Ref	Expression
Hypothesis	sbcgf.1	⊢ Ⅎ 𝑥 𝜑
Assertion	sbc19.21g	⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → [ 𝐴 / 𝑥 ] 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		sbcgf.1	⊢ Ⅎ 𝑥 𝜑
2		sbcimg	⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ ( [ 𝐴 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] 𝜓 ) ) )
3	1	sbcgf	⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑥 ] 𝜑 ↔ 𝜑 ) )
4	3	imbi1d	⊢ ( 𝐴 ∈ 𝑉 → ( ( [ 𝐴 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] 𝜓 ) ↔ ( 𝜑 → [ 𝐴 / 𝑥 ] 𝜓 ) ) )
5	2 4	bitrd	⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → [ 𝐴 / 𝑥 ] 𝜓 ) ) )