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Theorem infcllem 8378
Description: Lemma for infcl 8379, inflb 8380, infglb 8381, etc. (Contributed by AV, 3-Sep-2020.)
Hypotheses
Ref Expression
infcl.1 (𝜑𝑅 Or 𝐴)
infcl.2 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))
Assertion
Ref Expression
infcllem (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem infcllem
StepHypRef Expression
1 infcl.2 . 2 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))
2 vex 3198 . . . . . . . 8 𝑥 ∈ V
3 vex 3198 . . . . . . . 8 𝑦 ∈ V
42, 3brcnv 5294 . . . . . . 7 (𝑥𝑅𝑦𝑦𝑅𝑥)
54bicomi 214 . . . . . 6 (𝑦𝑅𝑥𝑥𝑅𝑦)
65notbii 310 . . . . 5 𝑦𝑅𝑥 ↔ ¬ 𝑥𝑅𝑦)
76ralbii 2977 . . . 4 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦𝐵 ¬ 𝑥𝑅𝑦)
83, 2brcnv 5294 . . . . . . 7 (𝑦𝑅𝑥𝑥𝑅𝑦)
98bicomi 214 . . . . . 6 (𝑥𝑅𝑦𝑦𝑅𝑥)
10 vex 3198 . . . . . . . . 9 𝑧 ∈ V
113, 10brcnv 5294 . . . . . . . 8 (𝑦𝑅𝑧𝑧𝑅𝑦)
1211bicomi 214 . . . . . . 7 (𝑧𝑅𝑦𝑦𝑅𝑧)
1312rexbii 3037 . . . . . 6 (∃𝑧𝐵 𝑧𝑅𝑦 ↔ ∃𝑧𝐵 𝑦𝑅𝑧)
149, 13imbi12i 340 . . . . 5 ((𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦) ↔ (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))
1514ralbii 2977 . . . 4 (∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦) ↔ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))
167, 15anbi12i 732 . . 3 ((∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)) ↔ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
1716rexbii 3037 . 2 (∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)) ↔ ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
181, 17sylib 208 1 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wral 2909  wrex 2910   class class class wbr 4644   Or wor 5024  ccnv 5103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704  df-cnv 5112
This theorem is referenced by:  infcl  8379  inflb  8380  infglb  8381  infglbb  8382  infiso  8398
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