Theorem exopxfr2 5698

Description: Transfer ordered-pair existence from/to single variable existence. (Contributed by NM, 26-Feb-2014.)

Hypothesis

Ref	Expression
exopxfr2.1	⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
exopxfr2	⊢ (Rel 𝐴 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦∃𝑧(⟨𝑦, 𝑧⟩ ∈ 𝐴 ∧ 𝜓)))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝜑,𝑦,𝑧   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦,𝑧)

Proof of Theorem exopxfr2

Step	Hyp	Ref	Expression
1		df-rel 5543	. . . . . . 7 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
2	1	biimpi 219	. . . . . 6 ⊢ (Rel 𝐴 → 𝐴 ⊆ (V × V))
3	2	sseld 3886	. . . . 5 ⊢ (Rel 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ (V × V)))
4	3	adantrd 495	. . . 4 ⊢ (Rel 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ (V × V)))
5	4	pm4.71rd 566	. . 3 ⊢ (Rel 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ (V × V) ∧ (𝑥 ∈ 𝐴 ∧ 𝜑))))
6	5	rexbidv2 3204	. 2 ⊢ (Rel 𝐴 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ (V × V)(𝑥 ∈ 𝐴 ∧ 𝜑)))
7		eleq1 2818	. . . 4 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ 𝐴 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
8		exopxfr2.1	. . . 4 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))
9	7, 8	anbi12d 634	. . 3 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (⟨𝑦, 𝑧⟩ ∈ 𝐴 ∧ 𝜓)))
10	9	exopxfr 5697	. 2 ⊢ (∃𝑥 ∈ (V × V)(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦∃𝑧(⟨𝑦, 𝑧⟩ ∈ 𝐴 ∧ 𝜓))
11	6, 10	bitrdi 290	1 ⊢ (Rel 𝐴 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦∃𝑧(⟨𝑦, 𝑧⟩ ∈ 𝐴 ∧ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543  ∃wex 1787   ∈ wcel 2112  ∃wrex 3052  Vcvv 3398   ⊆ wss 3853  ⟨cop 4533   × cxp 5534  Rel wrel 5541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-iun 4892  df-opab 5102  df-xp 5542  df-rel 5543

This theorem is referenced by:  dvhopellsm  38817