Theorem onfrALTlem4 44554

Description: Lemma for onfrALT 44560. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Distinct variable group:   𝑥,𝑎

Proof of Theorem onfrALTlem4

Step	Hyp	Ref	Expression
1		sbcan 3845	. 2 ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ ([𝑦 / 𝑥]𝑥 ∈ 𝑎 ∧ [𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅))
2		sbcel1v 3863	. . 3 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝑎 ↔ 𝑦 ∈ 𝑎)
3		vex 3483	. . . . 5 ⊢ 𝑦 ∈ V
4		sbceqg 4419	. . . . 5 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅ ↔ ⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = ⦋𝑦 / 𝑥⦌∅))
5	3, 4	ax-mp 5	. . . 4 ⊢ ([𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅ ↔ ⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = ⦋𝑦 / 𝑥⦌∅)
6		csbin 4449	. . . . . 6 ⊢ ⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = (⦋𝑦 / 𝑥⦌𝑎 ∩ ⦋𝑦 / 𝑥⦌𝑥)
7		csbconstg 3928	. . . . . . . 8 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑥⦌𝑎 = 𝑎)
8	3, 7	ax-mp 5	. . . . . . 7 ⊢ ⦋𝑦 / 𝑥⦌𝑎 = 𝑎
9		csbvarg 4441	. . . . . . . 8 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑥⦌𝑥 = 𝑦)
10	3, 9	ax-mp 5	. . . . . . 7 ⊢ ⦋𝑦 / 𝑥⦌𝑥 = 𝑦
11	8, 10	ineq12i 4227	. . . . . 6 ⊢ (⦋𝑦 / 𝑥⦌𝑎 ∩ ⦋𝑦 / 𝑥⦌𝑥) = (𝑎 ∩ 𝑦)
12	6, 11	eqtri 2764	. . . . 5 ⊢ ⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = (𝑎 ∩ 𝑦)
13		csb0 4417	. . . . 5 ⊢ ⦋𝑦 / 𝑥⦌∅ = ∅
14	12, 13	eqeq12i 2754	. . . 4 ⊢ (⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = ⦋𝑦 / 𝑥⦌∅ ↔ (𝑎 ∩ 𝑦) = ∅)
15	5, 14	bitri 275	. . 3 ⊢ ([𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅ ↔ (𝑎 ∩ 𝑦) = ∅)
16	2, 15	anbi12i 628	. 2 ⊢ (([𝑦 / 𝑥]𝑥 ∈ 𝑎 ∧ [𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
17	1, 16	bitri 275	1 ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1538   ∈ wcel 2107  Vcvv 3479  [wsbc 3792  ⦋csb 3909   ∩ cin 3963  ∅c0 4340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1541  df-fal 1551  df-ex 1778  df-nf 1782  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-rab 3435  df-v 3481  df-sbc 3793  df-csb 3910  df-dif 3967  df-in 3971  df-nul 4341

This theorem is referenced by:  onfrALTlem1  44559  onfrALTlem1VD  44901