Theorem exopxfr2 5805

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 5643	. . . . . . 7 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
2	1	biimpi 218	. . . . . 6 ⊢ (Rel 𝐴 → 𝐴 ⊆ (V × V))
3	2	sseld 3926	. . . . 5 ⊢ (Rel 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ (V × V)))
4	3	adantrd 494	. . . 4 ⊢ (Rel 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ (V × V)))
5	4	pm4.71rd 569	. . 3 ⊢ (Rel 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ (V × V) ∧ (𝑥 ∈ 𝐴 ∧ 𝜑))))
6	5	rexbidv2 3172	. 2 ⊢ (Rel 𝐴 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ (V × V)(𝑥 ∈ 𝐴 ∧ 𝜑)))
7		eleq1 2840	. . . 4 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ 𝐴 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
8		exopxfr2.1	. . . 4 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))
9	7, 8	anbi12d 640	. . 3 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (⟨𝑦, 𝑧⟩ ∈ 𝐴 ∧ 𝜓)))
10	9	exopxfr 5804	. 2 ⊢ (∃𝑥 ∈ (V × V)(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦∃𝑧(⟨𝑦, 𝑧⟩ ∈ 𝐴 ∧ 𝜓))
11	6, 10	bitrdi 289	1 ⊢ (Rel 𝐴 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦∃𝑧(⟨𝑦, 𝑧⟩ ∈ 𝐴 ∧ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1550  ∃wex 1789   ∈ wcel 2132  ∃wrex 3076  Vcvv 3444   ⊆ wss 3895  ⟨cop 4578   × cxp 5634  Rel wrel 5641

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-sep 5236  ax-pr 5380

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ral 3067  df-rex 3077  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-sn 4573  df-pr 4575  df-op 4579  df-iun 4941  df-opab 5153  df-xp 5642  df-rel 5643

This theorem is referenced by:  dvhopellsm  41679