Theorem onfrALTlem4VD 45329

Description: Virtual deduction proof of onfrALTlem4 44987. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. onfrALTlem4 44987 is onfrALTlem4VD 45329 without virtual deductions and was automatically derived from onfrALTlem4VD 45329.

1::	⊢ 𝑦 ∈ V
2:1:	⊢ ([𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅ ↔ ⦋ 𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = ⦋𝑦 / 𝑥⦌∅)
3:1:	⊢ ⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = (⦋𝑦 / 𝑥⦌ 𝑎 ∩ ⦋𝑦 / 𝑥⦌𝑥)
4:1:	⊢ ⦋𝑦 / 𝑥⦌𝑎 = 𝑎
5:1:	⊢ ⦋𝑦 / 𝑥⦌𝑥 = 𝑦
6:4,5:	⊢ (⦋𝑦 / 𝑥⦌𝑎 ∩ ⦋𝑦 / 𝑥⦌𝑥) = ( 𝑎 ∩ 𝑦)
7:3,6:	⊢ ⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = (𝑎 ∩ 𝑦)
8:1:	⊢ ⦋𝑦 / 𝑥⦌∅ = ∅
9:7,8:	⊢ (⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = ⦋𝑦 / 𝑥⦌ ∅ ↔ (𝑎 ∩ 𝑦) = ∅)
10:2,9:	⊢ ([𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅ ↔ (𝑎 ∩ 𝑦) = ∅)
11:1:	⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝑎 ↔ 𝑦 ∈ 𝑎)
12:11,10:	⊢ (([𝑦 / 𝑥]𝑥 ∈ 𝑎 ∧ [𝑦 / 𝑥]( 𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
13:1:	⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ ([𝑦 / 𝑥]𝑥 ∈ 𝑎 ∧ [𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅))
qed:13,12:	⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))

(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
onfrALTlem4VD	⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))

Distinct variable group:   𝑥,𝑎

Proof of Theorem onfrALTlem4VD

Step	Hyp	Ref	Expression
1		sbcan 3772	. 2 ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ ([𝑦 / 𝑥]𝑥 ∈ 𝑎 ∧ [𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅))
2		sbcel1v 3788	. . 3 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝑎 ↔ 𝑦 ∈ 𝑎)
3		sbceqg 4340	. . . . 5 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅ ↔ ⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = ⦋𝑦 / 𝑥⦌∅))
4	3	elv 3436	. . . 4 ⊢ ([𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅ ↔ ⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = ⦋𝑦 / 𝑥⦌∅)
5		csbin 4370	. . . . . 6 ⊢ ⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = (⦋𝑦 / 𝑥⦌𝑎 ∩ ⦋𝑦 / 𝑥⦌𝑥)
6		csbconstg 3850	. . . . . . . 8 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑥⦌𝑎 = 𝑎)
7	6	elv 3436	. . . . . . 7 ⊢ ⦋𝑦 / 𝑥⦌𝑎 = 𝑎
8		vex 3435	. . . . . . . 8 ⊢ 𝑦 ∈ V
9	8	csbvargi 4363	. . . . . . 7 ⊢ ⦋𝑦 / 𝑥⦌𝑥 = 𝑦
10	7, 9	ineq12i 4147	. . . . . 6 ⊢ (⦋𝑦 / 𝑥⦌𝑎 ∩ ⦋𝑦 / 𝑥⦌𝑥) = (𝑎 ∩ 𝑦)
11	5, 10	eqtri 2762	. . . . 5 ⊢ ⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = (𝑎 ∩ 𝑦)
12		csb0 4338	. . . . 5 ⊢ ⦋𝑦 / 𝑥⦌∅ = ∅
13	11, 12	eqeq12i 2757	. . . 4 ⊢ (⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = ⦋𝑦 / 𝑥⦌∅ ↔ (𝑎 ∩ 𝑦) = ∅)
14	4, 13	bitri 276	. . 3 ⊢ ([𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅ ↔ (𝑎 ∩ 𝑦) = ∅)
15	2, 14	anbi12i 634	. 2 ⊢ (([𝑦 / 𝑥]𝑥 ∈ 𝑎 ∧ [𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
16	1, 15	bitri 276	1 ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))

Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1547  Vcvv 3431  [wsbc 3723  ⦋csb 3831   ∩ cin 3882  ∅c0 4261

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-in 3890  df-nul 4262

This theorem is referenced by: (None)