Theorem brinxp2OLD 5427

Description: Obsolete version of brinxp2 5426 as of 18-Sep-2022. (Contributed by NM, 3-Mar-2007.) (Revised by Mario Carneiro, 26-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
brinxp2OLD	⊢ (𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵))

Proof of Theorem brinxp2OLD

Step	Hyp	Ref	Expression
1		brin 4938	. 2 ⊢ (𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵 ↔ (𝐴𝑅𝐵 ∧ 𝐴(𝐶 × 𝐷)𝐵))
2		ancom 454	. 2 ⊢ ((𝐴𝑅𝐵 ∧ 𝐴(𝐶 × 𝐷)𝐵) ↔ (𝐴(𝐶 × 𝐷)𝐵 ∧ 𝐴𝑅𝐵))
3		brxp 5401	. . . 4 ⊢ (𝐴(𝐶 × 𝐷)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷))
4	3	anbi1i 617	. . 3 ⊢ ((𝐴(𝐶 × 𝐷)𝐵 ∧ 𝐴𝑅𝐵) ↔ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ 𝐴𝑅𝐵))
5		df-3an 1073	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵) ↔ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ 𝐴𝑅𝐵))
6	4, 5	bitr4i 270	. 2 ⊢ ((𝐴(𝐶 × 𝐷)𝐵 ∧ 𝐴𝑅𝐵) ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵))
7	1, 2, 6	3bitri 289	1 ⊢ (𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵))

Syntax hints:   ↔ wb 198   ∧ wa 386   ∧ w3a 1071   ∈ wcel 2106   ∩ cin 3790   class class class wbr 4886   × cxp 5353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4887  df-opab 4949  df-xp 5361

This theorem is referenced by: (None)