Theorem exopxfr2 5779

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 5618	. . . . . . 7 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
2	1	biimpi 216	. . . . . 6 ⊢ (Rel 𝐴 → 𝐴 ⊆ (V × V))
3	2	sseld 3928	. . . . 5 ⊢ (Rel 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ (V × V)))
4	3	adantrd 491	. . . 4 ⊢ (Rel 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ (V × V)))
5	4	pm4.71rd 562	. . 3 ⊢ (Rel 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ (V × V) ∧ (𝑥 ∈ 𝐴 ∧ 𝜑))))
6	5	rexbidv2 3152	. 2 ⊢ (Rel 𝐴 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ (V × V)(𝑥 ∈ 𝐴 ∧ 𝜑)))
7		eleq1 2819	. . . 4 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ 𝐴 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
8		exopxfr2.1	. . . 4 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))
9	7, 8	anbi12d 632	. . 3 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (⟨𝑦, 𝑧⟩ ∈ 𝐴 ∧ 𝜓)))
10	9	exopxfr 5778	. 2 ⊢ (∃𝑥 ∈ (V × V)(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦∃𝑧(⟨𝑦, 𝑧⟩ ∈ 𝐴 ∧ 𝜓))
11	6, 10	bitrdi 287	1 ⊢ (Rel 𝐴 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦∃𝑧(⟨𝑦, 𝑧⟩ ∈ 𝐴 ∧ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2111  ∃wrex 3056  Vcvv 3436   ⊆ wss 3897  ⟨cop 4577   × cxp 5609  Rel wrel 5616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-iun 4938  df-opab 5149  df-xp 5617  df-rel 5618

This theorem is referenced by:  dvhopellsm  41156