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Metamath Proof Explorer

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 eqsbc2 ⊢ ( 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 ) ) )