Theorem sbcoreleleq

Description: Substitution of a setvar variable for another setvar variable in a 3-conjunct formula. Derived automatically from sbcoreleleqVD . (Contributed by Alan Sare, 31-Dec-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	sbcoreleleq	⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑦 ] ( 𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦 ) ↔ ( 𝑥 ∈ 𝐴 ∨ 𝐴 ∈ 𝑥 ∨ 𝑥 = 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		sbc3or	⊢ ( [ 𝐴 / 𝑦 ] ( 𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦 ) ↔ ( [ 𝐴 / 𝑦 ] 𝑥 ∈ 𝑦 ∨ [ 𝐴 / 𝑦 ] 𝑦 ∈ 𝑥 ∨ [ 𝐴 / 𝑦 ] 𝑥 = 𝑦 ) )
2		sbcel2gv	⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑦 ] 𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝐴 ) )
3		sbcel1v	⊢ ( [ 𝐴 / 𝑦 ] 𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥 )
4	3	a1i	⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑦 ] 𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥 ) )
5		eqsbc3r	⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑦 ] 𝑥 = 𝑦 ↔ 𝑥 = 𝐴 ) )
6		3orbi123	⊢ ( ( ( [ 𝐴 / 𝑦 ] 𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝐴 ) ∧ ( [ 𝐴 / 𝑦 ] 𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥 ) ∧ ( [ 𝐴 / 𝑦 ] 𝑥 = 𝑦 ↔ 𝑥 = 𝐴 ) ) → ( ( [ 𝐴 / 𝑦 ] 𝑥 ∈ 𝑦 ∨ [ 𝐴 / 𝑦 ] 𝑦 ∈ 𝑥 ∨ [ 𝐴 / 𝑦 ] 𝑥 = 𝑦 ) ↔ ( 𝑥 ∈ 𝐴 ∨ 𝐴 ∈ 𝑥 ∨ 𝑥 = 𝐴 ) ) )
7	6	3impexpbicomi	⊢ ( ( [ 𝐴 / 𝑦 ] 𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝐴 ) → ( ( [ 𝐴 / 𝑦 ] 𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥 ) → ( ( [ 𝐴 / 𝑦 ] 𝑥 = 𝑦 ↔ 𝑥 = 𝐴 ) → ( ( 𝑥 ∈ 𝐴 ∨ 𝐴 ∈ 𝑥 ∨ 𝑥 = 𝐴 ) ↔ ( [ 𝐴 / 𝑦 ] 𝑥 ∈ 𝑦 ∨ [ 𝐴 / 𝑦 ] 𝑦 ∈ 𝑥 ∨ [ 𝐴 / 𝑦 ] 𝑥 = 𝑦 ) ) ) ) )
8	2 4 5 7	syl3c	⊢ ( 𝐴 ∈ 𝑉 → ( ( 𝑥 ∈ 𝐴 ∨ 𝐴 ∈ 𝑥 ∨ 𝑥 = 𝐴 ) ↔ ( [ 𝐴 / 𝑦 ] 𝑥 ∈ 𝑦 ∨ [ 𝐴 / 𝑦 ] 𝑦 ∈ 𝑥 ∨ [ 𝐴 / 𝑦 ] 𝑥 = 𝑦 ) ) )
9	1 8	bitr4id	⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑦 ] ( 𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦 ) ↔ ( 𝑥 ∈ 𝐴 ∨ 𝐴 ∈ 𝑥 ∨ 𝑥 = 𝐴 ) ) )