Theorem sbcbig

Description: Distribution of class substitution over biconditional. (Contributed by Raph Levien, 10-Apr-2004)

	Ref	Expression
Assertion	sbcbig	⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑥 ] ( 𝜑 ↔ 𝜓 ) ↔ ( [ 𝐴 / 𝑥 ] 𝜑 ↔ [ 𝐴 / 𝑥 ] 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		dfsbcq2	⊢ ( 𝑦 = 𝐴 → ( [ 𝑦 / 𝑥 ] ( 𝜑 ↔ 𝜓 ) ↔ [ 𝐴 / 𝑥 ] ( 𝜑 ↔ 𝜓 ) ) )
2		dfsbcq2	⊢ ( 𝑦 = 𝐴 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝐴 / 𝑥 ] 𝜑 ) )
3		dfsbcq2	⊢ ( 𝑦 = 𝐴 → ( [ 𝑦 / 𝑥 ] 𝜓 ↔ [ 𝐴 / 𝑥 ] 𝜓 ) )
4	2 3	bibi12d	⊢ ( 𝑦 = 𝐴 → ( ( [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜓 ) ↔ ( [ 𝐴 / 𝑥 ] 𝜑 ↔ [ 𝐴 / 𝑥 ] 𝜓 ) ) )
5		sbbi	⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 ↔ 𝜓 ) ↔ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜓 ) )
6	1 4 5	vtoclbg	⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑥 ] ( 𝜑 ↔ 𝜓 ) ↔ ( [ 𝐴 / 𝑥 ] 𝜑 ↔ [ 𝐴 / 𝑥 ] 𝜓 ) ) )