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Theorem elovmpo 7678
Description: Utility lemma for two-parameter classes.

EDITORIAL: can simplify islmhm 21026. (Contributed by Stefan O'Rear, 21-Jan-2015.)

Hypotheses
Ref Expression
elovmpo.d 𝐷 = (𝑎𝐴, 𝑏𝐵𝐶)
elovmpo.c 𝐶 ∈ V
elovmpo.e ((𝑎 = 𝑋𝑏 = 𝑌) → 𝐶 = 𝐸)
Assertion
Ref Expression
elovmpo (𝐹 ∈ (𝑋𝐷𝑌) ↔ (𝑋𝐴𝑌𝐵𝐹𝐸))
Distinct variable groups:   𝐴,𝑎,𝑏   𝐵,𝑎,𝑏   𝐸,𝑎,𝑏   𝐹,𝑎,𝑏   𝑋,𝑎,𝑏   𝑌,𝑎,𝑏
Allowed substitution hints:   𝐶(𝑎,𝑏)   𝐷(𝑎,𝑏)

Proof of Theorem elovmpo
StepHypRef Expression
1 elovmpo.d . . . 4 𝐷 = (𝑎𝐴, 𝑏𝐵𝐶)
21elmpocl 7674 . . 3 (𝐹 ∈ (𝑋𝐷𝑌) → (𝑋𝐴𝑌𝐵))
3 elovmpo.c . . . . . . 7 𝐶 ∈ V
43gen2 1796 . . . . . 6 𝑎𝑏 𝐶 ∈ V
5 elovmpo.e . . . . . . . 8 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝐶 = 𝐸)
65eleq1d 2826 . . . . . . 7 ((𝑎 = 𝑋𝑏 = 𝑌) → (𝐶 ∈ V ↔ 𝐸 ∈ V))
76spc2gv 3600 . . . . . 6 ((𝑋𝐴𝑌𝐵) → (∀𝑎𝑏 𝐶 ∈ V → 𝐸 ∈ V))
84, 7mpi 20 . . . . 5 ((𝑋𝐴𝑌𝐵) → 𝐸 ∈ V)
95, 1ovmpoga 7587 . . . . 5 ((𝑋𝐴𝑌𝐵𝐸 ∈ V) → (𝑋𝐷𝑌) = 𝐸)
108, 9mpd3an3 1464 . . . 4 ((𝑋𝐴𝑌𝐵) → (𝑋𝐷𝑌) = 𝐸)
1110eleq2d 2827 . . 3 ((𝑋𝐴𝑌𝐵) → (𝐹 ∈ (𝑋𝐷𝑌) ↔ 𝐹𝐸))
122, 11biadanii 822 . 2 (𝐹 ∈ (𝑋𝐷𝑌) ↔ ((𝑋𝐴𝑌𝐵) ∧ 𝐹𝐸))
13 df-3an 1089 . 2 ((𝑋𝐴𝑌𝐵𝐹𝐸) ↔ ((𝑋𝐴𝑌𝐵) ∧ 𝐹𝐸))
1412, 13bitr4i 278 1 (𝐹 ∈ (𝑋𝐷𝑌) ↔ (𝑋𝐴𝑌𝐵𝐹𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087  wal 1538   = wceq 1540  wcel 2108  Vcvv 3480  (class class class)co 7431  cmpo 7433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436
This theorem is referenced by:  isgim  19280  oppglsm  19660  islmim  21061  sn-isghm  42683
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