Theorem onfrALTlem4

Description: Lemma for onfrALT . (Contributed by Alan Sare, 22-Jul-2012) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	onfrALTlem4	⊢ ( [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) ↔ ( 𝑦 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑦 ) = ∅ ) )

Proof

Step	Hyp	Ref	Expression
1		sbcan	⊢ ( [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) ↔ ( [ 𝑦 / 𝑥 ] 𝑥 ∈ 𝑎 ∧ [ 𝑦 / 𝑥 ] ( 𝑎 ∩ 𝑥 ) = ∅ ) )
2		sbcel1v	⊢ ( [ 𝑦 / 𝑥 ] 𝑥 ∈ 𝑎 ↔ 𝑦 ∈ 𝑎 )
3		vex	⊢ 𝑦 ∈ V
4		sbceqg	⊢ ( 𝑦 ∈ V → ( [ 𝑦 / 𝑥 ] ( 𝑎 ∩ 𝑥 ) = ∅ ↔ ⦋ 𝑦 / 𝑥 ⦌ ( 𝑎 ∩ 𝑥 ) = ⦋ 𝑦 / 𝑥 ⦌ ∅ ) )
5	3 4	ax-mp	⊢ ( [ 𝑦 / 𝑥 ] ( 𝑎 ∩ 𝑥 ) = ∅ ↔ ⦋ 𝑦 / 𝑥 ⦌ ( 𝑎 ∩ 𝑥 ) = ⦋ 𝑦 / 𝑥 ⦌ ∅ )
6		csbin	⊢ ⦋ 𝑦 / 𝑥 ⦌ ( 𝑎 ∩ 𝑥 ) = ( ⦋ 𝑦 / 𝑥 ⦌ 𝑎 ∩ ⦋ 𝑦 / 𝑥 ⦌ 𝑥 )
7		csbconstg	⊢ ( 𝑦 ∈ V → ⦋ 𝑦 / 𝑥 ⦌ 𝑎 = 𝑎 )
8	3 7	ax-mp	⊢ ⦋ 𝑦 / 𝑥 ⦌ 𝑎 = 𝑎
9		csbvarg	⊢ ( 𝑦 ∈ V → ⦋ 𝑦 / 𝑥 ⦌ 𝑥 = 𝑦 )
10	3 9	ax-mp	⊢ ⦋ 𝑦 / 𝑥 ⦌ 𝑥 = 𝑦
11	8 10	ineq12i	⊢ ( ⦋ 𝑦 / 𝑥 ⦌ 𝑎 ∩ ⦋ 𝑦 / 𝑥 ⦌ 𝑥 ) = ( 𝑎 ∩ 𝑦 )
12	6 11	eqtri	⊢ ⦋ 𝑦 / 𝑥 ⦌ ( 𝑎 ∩ 𝑥 ) = ( 𝑎 ∩ 𝑦 )
13		csb0	⊢ ⦋ 𝑦 / 𝑥 ⦌ ∅ = ∅
14	12 13	eqeq12i	⊢ ( ⦋ 𝑦 / 𝑥 ⦌ ( 𝑎 ∩ 𝑥 ) = ⦋ 𝑦 / 𝑥 ⦌ ∅ ↔ ( 𝑎 ∩ 𝑦 ) = ∅ )
15	5 14	bitri	⊢ ( [ 𝑦 / 𝑥 ] ( 𝑎 ∩ 𝑥 ) = ∅ ↔ ( 𝑎 ∩ 𝑦 ) = ∅ )
16	2 15	anbi12i	⊢ ( ( [ 𝑦 / 𝑥 ] 𝑥 ∈ 𝑎 ∧ [ 𝑦 / 𝑥 ] ( 𝑎 ∩ 𝑥 ) = ∅ ) ↔ ( 𝑦 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑦 ) = ∅ ) )
17	1 16	bitri	⊢ ( [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) ↔ ( 𝑦 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑦 ) = ∅ ) )