Theorem sbc4rexgOLD

Description: Exchange a substitution with four existentials. (Contributed by Stefan O'Rear, 11-Oct-2014) Obsolete as of 24-Aug-2018. Use csbov123 instead. (New usage is discouraged.) (Proof modification is discouraged.)

	Ref	Expression
Assertion	sbc4rexgOLD	⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑎 ] ∃ 𝑏 ∈ 𝐵 ∃ 𝑐 ∈ 𝐶 ∃ 𝑑 ∈ 𝐷 ∃ 𝑒 ∈ 𝐸 𝜑 ↔ ∃ 𝑏 ∈ 𝐵 ∃ 𝑐 ∈ 𝐶 ∃ 𝑑 ∈ 𝐷 ∃ 𝑒 ∈ 𝐸 [ 𝐴 / 𝑎 ] 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		elex	⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ V )
2		sbc2rexgOLD	⊢ ( 𝐴 ∈ V → ( [ 𝐴 / 𝑎 ] ∃ 𝑏 ∈ 𝐵 ∃ 𝑐 ∈ 𝐶 ∃ 𝑑 ∈ 𝐷 ∃ 𝑒 ∈ 𝐸 𝜑 ↔ ∃ 𝑏 ∈ 𝐵 ∃ 𝑐 ∈ 𝐶 [ 𝐴 / 𝑎 ] ∃ 𝑑 ∈ 𝐷 ∃ 𝑒 ∈ 𝐸 𝜑 ) )
3		sbc2rexgOLD	⊢ ( 𝐴 ∈ V → ( [ 𝐴 / 𝑎 ] ∃ 𝑑 ∈ 𝐷 ∃ 𝑒 ∈ 𝐸 𝜑 ↔ ∃ 𝑑 ∈ 𝐷 ∃ 𝑒 ∈ 𝐸 [ 𝐴 / 𝑎 ] 𝜑 ) )
4	3	2rexbidv	⊢ ( 𝐴 ∈ V → ( ∃ 𝑏 ∈ 𝐵 ∃ 𝑐 ∈ 𝐶 [ 𝐴 / 𝑎 ] ∃ 𝑑 ∈ 𝐷 ∃ 𝑒 ∈ 𝐸 𝜑 ↔ ∃ 𝑏 ∈ 𝐵 ∃ 𝑐 ∈ 𝐶 ∃ 𝑑 ∈ 𝐷 ∃ 𝑒 ∈ 𝐸 [ 𝐴 / 𝑎 ] 𝜑 ) )
5	2 4	bitrd	⊢ ( 𝐴 ∈ V → ( [ 𝐴 / 𝑎 ] ∃ 𝑏 ∈ 𝐵 ∃ 𝑐 ∈ 𝐶 ∃ 𝑑 ∈ 𝐷 ∃ 𝑒 ∈ 𝐸 𝜑 ↔ ∃ 𝑏 ∈ 𝐵 ∃ 𝑐 ∈ 𝐶 ∃ 𝑑 ∈ 𝐷 ∃ 𝑒 ∈ 𝐸 [ 𝐴 / 𝑎 ] 𝜑 ) )
6	1 5	syl	⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑎 ] ∃ 𝑏 ∈ 𝐵 ∃ 𝑐 ∈ 𝐶 ∃ 𝑑 ∈ 𝐷 ∃ 𝑒 ∈ 𝐸 𝜑 ↔ ∃ 𝑏 ∈ 𝐵 ∃ 𝑐 ∈ 𝐶 ∃ 𝑑 ∈ 𝐷 ∃ 𝑒 ∈ 𝐸 [ 𝐴 / 𝑎 ] 𝜑 ) )