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Theorem prtlem10 33665
Description: Lemma for prter3 33682. (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 477 . . . . 5 (( Er 𝐴𝑧𝐴) → 𝑧𝐴)
2 simpl 473 . . . . . 6 (( Er 𝐴𝑧𝐴) → Er 𝐴)
32, 1erref 7714 . . . . 5 (( Er 𝐴𝑧𝐴) → 𝑧 𝑧)
4 breq1 4621 . . . . . . . 8 (𝑣 = 𝑧 → (𝑣 𝑧𝑧 𝑧))
5 breq1 4621 . . . . . . . 8 (𝑣 = 𝑧 → (𝑣 𝑤𝑧 𝑤))
64, 5anbi12d 746 . . . . . . 7 (𝑣 = 𝑧 → ((𝑣 𝑧𝑣 𝑤) ↔ (𝑧 𝑧𝑧 𝑤)))
76rspcev 3298 . . . . . 6 ((𝑧𝐴 ∧ (𝑧 𝑧𝑧 𝑤)) → ∃𝑣𝐴 (𝑣 𝑧𝑣 𝑤))
87expr 642 . . . . 5 ((𝑧𝐴𝑧 𝑧) → (𝑧 𝑤 → ∃𝑣𝐴 (𝑣 𝑧𝑣 𝑤)))
91, 3, 8syl2anc 692 . . . 4 (( Er 𝐴𝑧𝐴) → (𝑧 𝑤 → ∃𝑣𝐴 (𝑣 𝑧𝑣 𝑤)))
10 simplll 797 . . . . . . 7 (((( Er 𝐴𝑧𝐴) ∧ 𝑣𝐴) ∧ (𝑣 𝑧𝑣 𝑤)) → Er 𝐴)
11 simprl 793 . . . . . . 7 (((( Er 𝐴𝑧𝐴) ∧ 𝑣𝐴) ∧ (𝑣 𝑧𝑣 𝑤)) → 𝑣 𝑧)
12 simprr 795 . . . . . . 7 (((( Er 𝐴𝑧𝐴) ∧ 𝑣𝐴) ∧ (𝑣 𝑧𝑣 𝑤)) → 𝑣 𝑤)
1310, 11, 12ertr3d 7712 . . . . . 6 (((( Er 𝐴𝑧𝐴) ∧ 𝑣𝐴) ∧ (𝑣 𝑧𝑣 𝑤)) → 𝑧 𝑤)
1413ex 450 . . . . 5 ((( Er 𝐴𝑧𝐴) ∧ 𝑣𝐴) → ((𝑣 𝑧𝑣 𝑤) → 𝑧 𝑤))
1514rexlimdva 3025 . . . 4 (( Er 𝐴𝑧𝐴) → (∃𝑣𝐴 (𝑣 𝑧𝑣 𝑤) → 𝑧 𝑤))
169, 15impbid 202 . . 3 (( Er 𝐴𝑧𝐴) → (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑣 𝑧𝑣 𝑤)))
17 vex 3192 . . . . . 6 𝑧 ∈ V
18 vex 3192 . . . . . 6 𝑣 ∈ V
1917, 18elec 7738 . . . . 5 (𝑧 ∈ [𝑣] 𝑣 𝑧)
20 vex 3192 . . . . . 6 𝑤 ∈ V
2120, 18elec 7738 . . . . 5 (𝑤 ∈ [𝑣] 𝑣 𝑤)
2219, 21anbi12i 732 . . . 4 ((𝑧 ∈ [𝑣] 𝑤 ∈ [𝑣] ) ↔ (𝑣 𝑧𝑣 𝑤))
2322rexbii 3035 . . 3 (∃𝑣𝐴 (𝑧 ∈ [𝑣] 𝑤 ∈ [𝑣] ) ↔ ∃𝑣𝐴 (𝑣 𝑧𝑣 𝑤))
2416, 23syl6bbr 278 . 2 (( Er 𝐴𝑧𝐴) → (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑧 ∈ [𝑣] 𝑤 ∈ [𝑣] )))
2524ex 450 1 ( Er 𝐴 → (𝑧𝐴 → (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑧 ∈ [𝑣] 𝑤 ∈ [𝑣] ))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wcel 1987  wrex 2908   class class class wbr 4618   Er wer 7691  [cec 7692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-er 7694  df-ec 7696
This theorem is referenced by: (None)
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