Theorem prtlem13 39367

Description: Lemma for prter1 39378, prter2 39380, prter3 39381 and prtex 39379. (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypothesis

Ref	Expression
prtlem13.1	⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}

Assertion

Ref	Expression
prtlem13	⊢ (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))

Distinct variable groups:   𝑣,𝑢,𝑥,𝑦,𝐴   𝑤,𝑣,𝑥,𝑦   𝑧,𝑣,𝑥,𝑦

Allowed substitution hints:   𝐴(𝑧,𝑤)   ∼ (𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)

Proof of Theorem prtlem13

Step	Hyp	Ref	Expression
1		vex 3436	. 2 ⊢ 𝑧 ∈ V
2		vex 3436	. 2 ⊢ 𝑤 ∈ V
3		elequ2 2134	. . . . 5 ⊢ (𝑢 = 𝑣 → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ 𝑣))
4		elequ2 2134	. . . . 5 ⊢ (𝑢 = 𝑣 → (𝑦 ∈ 𝑢 ↔ 𝑦 ∈ 𝑣))
5	3, 4	anbi12d 638	. . . 4 ⊢ (𝑢 = 𝑣 → ((𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢) ↔ (𝑥 ∈ 𝑣 ∧ 𝑦 ∈ 𝑣)))
6	5	cbvrexvw 3219	. . 3 ⊢ (∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢) ↔ ∃𝑣 ∈ 𝐴 (𝑥 ∈ 𝑣 ∧ 𝑦 ∈ 𝑣))
7		elequ1 2126	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝑣 ↔ 𝑧 ∈ 𝑣))
8		elequ1 2126	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝑣 ↔ 𝑤 ∈ 𝑣))
9	7, 8	bi2anan9 644	. . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ∈ 𝑣 ∧ 𝑦 ∈ 𝑣) ↔ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
10	9	rexbidv 3164	. . 3 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (∃𝑣 ∈ 𝐴 (𝑥 ∈ 𝑣 ∧ 𝑦 ∈ 𝑣) ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
11	6, 10	bitrid 284	. 2 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢) ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
12		prtlem13.1	. 2 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}
13	1, 2, 11, 12	braba 5486	1 ⊢ (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))

Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1547  ∃wrex 3064   class class class wbr 5079  {copab 5141

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-ext 2712  ax-sep 5225  ax-pr 5369

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-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142

This theorem is referenced by:  prtlem16  39368  prtlem18  39376  prter1  39378  prter3  39381