Theorem prtlem13 35485

Description: Lemma for prter1 35496, prter2 35498, prter3 35499 and prtex 35497. (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 3435	. 2 ⊢ 𝑧 ∈ V
2		vex 3435	. 2 ⊢ 𝑤 ∈ V
3		elequ2 2094	. . . . 5 ⊢ (𝑢 = 𝑣 → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ 𝑣))
4		elequ2 2094	. . . . 5 ⊢ (𝑢 = 𝑣 → (𝑦 ∈ 𝑢 ↔ 𝑦 ∈ 𝑣))
5	3, 4	anbi12d 630	. . . 4 ⊢ (𝑢 = 𝑣 → ((𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢) ↔ (𝑥 ∈ 𝑣 ∧ 𝑦 ∈ 𝑣)))
6	5	cbvrexv 3401	. . 3 ⊢ (∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢) ↔ ∃𝑣 ∈ 𝐴 (𝑥 ∈ 𝑣 ∧ 𝑦 ∈ 𝑣))
7		elequ1 2086	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝑣 ↔ 𝑧 ∈ 𝑣))
8		elequ1 2086	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝑣 ↔ 𝑤 ∈ 𝑣))
9	7, 8	bi2anan9 635	. . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ∈ 𝑣 ∧ 𝑦 ∈ 𝑣) ↔ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
10	9	rexbidv 3257	. . 3 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (∃𝑣 ∈ 𝐴 (𝑥 ∈ 𝑣 ∧ 𝑦 ∈ 𝑣) ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
11	6, 10	syl5bb 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 5306	1 ⊢ (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))

Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1520  ∃wrex 3104   class class class wbr 4956  {copab 5018

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-sep 5088  ax-nul 5095  ax-pr 5214

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3434  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-op 4473  df-br 4957  df-opab 5019

This theorem is referenced by:  prtlem16  35486  prtlem18  35494  prter1  35496  prter3  35499