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Theorem tz7.2 5058
 Description: Similar to Theorem 7.2 of [TakeutiZaring] p. 35, of except that the Axiom of Regularity is not required due to antecedent E Fr 𝐴. (Contributed by NM, 4-May-1994.)
Assertion
Ref Expression
tz7.2 ((Tr 𝐴 ∧ E Fr 𝐴𝐵𝐴) → (𝐵𝐴𝐵𝐴))

Proof of Theorem tz7.2
StepHypRef Expression
1 trss 4721 . . 3 (Tr 𝐴 → (𝐵𝐴𝐵𝐴))
2 efrirr 5055 . . . . 5 ( E Fr 𝐴 → ¬ 𝐴𝐴)
3 eleq1 2686 . . . . . 6 (𝐵 = 𝐴 → (𝐵𝐴𝐴𝐴))
43notbid 308 . . . . 5 (𝐵 = 𝐴 → (¬ 𝐵𝐴 ↔ ¬ 𝐴𝐴))
52, 4syl5ibrcom 237 . . . 4 ( E Fr 𝐴 → (𝐵 = 𝐴 → ¬ 𝐵𝐴))
65necon2ad 2805 . . 3 ( E Fr 𝐴 → (𝐵𝐴𝐵𝐴))
71, 6anim12ii 593 . 2 ((Tr 𝐴 ∧ E Fr 𝐴) → (𝐵𝐴 → (𝐵𝐴𝐵𝐴)))
873impia 1258 1 ((Tr 𝐴 ∧ E Fr 𝐴𝐵𝐴) → (𝐵𝐴𝐵𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987   ≠ wne 2790   ⊆ wss 3555  Tr wtr 4712   E cep 4983   Fr wfr 5030 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 4741  ax-nul 4749  ax-pr 4867 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-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-tr 4713  df-eprel 4985  df-fr 5033 This theorem is referenced by:  tz7.7  5708  trelpss  38138
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