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Theorem exopxfr2 5173
Description: Transfer ordered-pair existence from/to single variable existence. (Contributed by NM, 26-Feb-2014.)
Hypothesis
Ref Expression
exopxfr2.1 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))
Assertion
Ref Expression
exopxfr2 (Rel 𝐴 → (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝑧(⟨𝑦, 𝑧⟩ ∈ 𝐴𝜓)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝜑,𝑦,𝑧   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦,𝑧)

Proof of Theorem exopxfr2
StepHypRef Expression
1 df-rel 5032 . . . . . . 7 (Rel 𝐴𝐴 ⊆ (V × V))
21biimpi 204 . . . . . 6 (Rel 𝐴𝐴 ⊆ (V × V))
32sseld 3563 . . . . 5 (Rel 𝐴 → (𝑥𝐴𝑥 ∈ (V × V)))
43adantrd 482 . . . 4 (Rel 𝐴 → ((𝑥𝐴𝜑) → 𝑥 ∈ (V × V)))
54pm4.71rd 664 . . 3 (Rel 𝐴 → ((𝑥𝐴𝜑) ↔ (𝑥 ∈ (V × V) ∧ (𝑥𝐴𝜑))))
65rexbidv2 3026 . 2 (Rel 𝐴 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥 ∈ (V × V)(𝑥𝐴𝜑)))
7 eleq1 2672 . . . 4 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥𝐴 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
8 exopxfr2.1 . . . 4 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))
97, 8anbi12d 742 . . 3 (𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑥𝐴𝜑) ↔ (⟨𝑦, 𝑧⟩ ∈ 𝐴𝜓)))
109exopxfr 5172 . 2 (∃𝑥 ∈ (V × V)(𝑥𝐴𝜑) ↔ ∃𝑦𝑧(⟨𝑦, 𝑧⟩ ∈ 𝐴𝜓))
116, 10syl6bb 274 1 (Rel 𝐴 → (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝑧(⟨𝑦, 𝑧⟩ ∈ 𝐴𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wex 1694  wcel 1976  wrex 2893  Vcvv 3169  wss 3536  cop 4127   × cxp 5023  Rel wrel 5030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pr 4825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-op 4128  df-iun 4448  df-opab 4635  df-xp 5031  df-rel 5032
This theorem is referenced by:  dvhopellsm  35224
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