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Theorem prtlem10 37735
Description: Lemma for prter3 37752. (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
Assertion
Ref Expression
prtlem10 ( Er 𝐴 → (𝑧𝐴 → (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑧 ∈ [𝑣] 𝑤 ∈ [𝑣] ))))
Distinct variable groups:   𝑤,𝑣   𝑧,𝑣   𝑣,𝐴   𝑣,
Allowed substitution hints:   𝐴(𝑧,𝑤)   (𝑧,𝑤)

Proof of Theorem prtlem10
StepHypRef Expression
1 simpr 486 . . . . 5 (( Er 𝐴𝑧𝐴) → 𝑧𝐴)
2 simpl 484 . . . . . 6 (( Er 𝐴𝑧𝐴) → Er 𝐴)
32, 1erref 8723 . . . . 5 (( Er 𝐴𝑧𝐴) → 𝑧 𝑧)
4 breq1 5152 . . . . . . . 8 (𝑣 = 𝑧 → (𝑣 𝑧𝑧 𝑧))
5 breq1 5152 . . . . . . . 8 (𝑣 = 𝑧 → (𝑣 𝑤𝑧 𝑤))
64, 5anbi12d 632 . . . . . . 7 (𝑣 = 𝑧 → ((𝑣 𝑧𝑣 𝑤) ↔ (𝑧 𝑧𝑧 𝑤)))
76rspcev 3613 . . . . . 6 ((𝑧𝐴 ∧ (𝑧 𝑧𝑧 𝑤)) → ∃𝑣𝐴 (𝑣 𝑧𝑣 𝑤))
87expr 458 . . . . 5 ((𝑧𝐴𝑧 𝑧) → (𝑧 𝑤 → ∃𝑣𝐴 (𝑣 𝑧𝑣 𝑤)))
91, 3, 8syl2anc 585 . . . 4 (( Er 𝐴𝑧𝐴) → (𝑧 𝑤 → ∃𝑣𝐴 (𝑣 𝑧𝑣 𝑤)))
10 simplll 774 . . . . . 6 (((( Er 𝐴𝑧𝐴) ∧ 𝑣𝐴) ∧ (𝑣 𝑧𝑣 𝑤)) → Er 𝐴)
11 simprl 770 . . . . . 6 (((( Er 𝐴𝑧𝐴) ∧ 𝑣𝐴) ∧ (𝑣 𝑧𝑣 𝑤)) → 𝑣 𝑧)
12 simprr 772 . . . . . 6 (((( Er 𝐴𝑧𝐴) ∧ 𝑣𝐴) ∧ (𝑣 𝑧𝑣 𝑤)) → 𝑣 𝑤)
1310, 11, 12ertr3d 8721 . . . . 5 (((( Er 𝐴𝑧𝐴) ∧ 𝑣𝐴) ∧ (𝑣 𝑧𝑣 𝑤)) → 𝑧 𝑤)
1413rexlimdva2 3158 . . . 4 (( Er 𝐴𝑧𝐴) → (∃𝑣𝐴 (𝑣 𝑧𝑣 𝑤) → 𝑧 𝑤))
159, 14impbid 211 . . 3 (( Er 𝐴𝑧𝐴) → (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑣 𝑧𝑣 𝑤)))
16 vex 3479 . . . . . 6 𝑧 ∈ V
17 vex 3479 . . . . . 6 𝑣 ∈ V
1816, 17elec 8747 . . . . 5 (𝑧 ∈ [𝑣] 𝑣 𝑧)
19 vex 3479 . . . . . 6 𝑤 ∈ V
2019, 17elec 8747 . . . . 5 (𝑤 ∈ [𝑣] 𝑣 𝑤)
2118, 20anbi12i 628 . . . 4 ((𝑧 ∈ [𝑣] 𝑤 ∈ [𝑣] ) ↔ (𝑣 𝑧𝑣 𝑤))
2221rexbii 3095 . . 3 (∃𝑣𝐴 (𝑧 ∈ [𝑣] 𝑤 ∈ [𝑣] ) ↔ ∃𝑣𝐴 (𝑣 𝑧𝑣 𝑤))
2315, 22bitr4di 289 . 2 (( Er 𝐴𝑧𝐴) → (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑧 ∈ [𝑣] 𝑤 ∈ [𝑣] )))
2423ex 414 1 ( Er 𝐴 → (𝑧𝐴 → (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑧 ∈ [𝑣] 𝑤 ∈ [𝑣] ))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wcel 2107  wrex 3071   class class class wbr 5149   Er wer 8700  [cec 8701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-er 8703  df-ec 8705
This theorem is referenced by: (None)
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