Theorem sbcrexgOLD

Description: Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005) (Proof shortened by Andrew Salmon, 29-Jun-2011) Obsolete as of 18-Aug-2018. Use sbcrex instead. (New usage is discouraged.) (Proof modification is discouraged.)

	Ref	Expression
Assertion	sbcrexgOLD	⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑥 ] ∃ 𝑦 ∈ 𝐵 𝜑 ↔ ∃ 𝑦 ∈ 𝐵 [ 𝐴 / 𝑥 ] 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		dfsbcq2	⊢ ( 𝑧 = 𝐴 → ( [ 𝑧 / 𝑥 ] ∃ 𝑦 ∈ 𝐵 𝜑 ↔ [ 𝐴 / 𝑥 ] ∃ 𝑦 ∈ 𝐵 𝜑 ) )
2		dfsbcq2	⊢ ( 𝑧 = 𝐴 → ( [ 𝑧 / 𝑥 ] 𝜑 ↔ [ 𝐴 / 𝑥 ] 𝜑 ) )
3	2	rexbidv	⊢ ( 𝑧 = 𝐴 → ( ∃ 𝑦 ∈ 𝐵 [ 𝑧 / 𝑥 ] 𝜑 ↔ ∃ 𝑦 ∈ 𝐵 [ 𝐴 / 𝑥 ] 𝜑 ) )
4		nfcv	⊢ Ⅎ 𝑥 𝐵
5		nfs1v	⊢ Ⅎ 𝑥 [ 𝑧 / 𝑥 ] 𝜑
6	4 5	nfrex	⊢ Ⅎ 𝑥 ∃ 𝑦 ∈ 𝐵 [ 𝑧 / 𝑥 ] 𝜑
7		sbequ12	⊢ ( 𝑥 = 𝑧 → ( 𝜑 ↔ [ 𝑧 / 𝑥 ] 𝜑 ) )
8	7	rexbidv	⊢ ( 𝑥 = 𝑧 → ( ∃ 𝑦 ∈ 𝐵 𝜑 ↔ ∃ 𝑦 ∈ 𝐵 [ 𝑧 / 𝑥 ] 𝜑 ) )
9	6 8	sbie	⊢ ( [ 𝑧 / 𝑥 ] ∃ 𝑦 ∈ 𝐵 𝜑 ↔ ∃ 𝑦 ∈ 𝐵 [ 𝑧 / 𝑥 ] 𝜑 )
10	1 3 9	vtoclbg	⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑥 ] ∃ 𝑦 ∈ 𝐵 𝜑 ↔ ∃ 𝑦 ∈ 𝐵 [ 𝐴 / 𝑥 ] 𝜑 ) )