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Theorem tz7.2 4116
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 3888 . . 3 (Tr 𝐴 → (𝐵𝐴𝐵𝐴))
2 efrirr 4115 . . . . 5 ( E Fr 𝐴 → ¬ 𝐴𝐴)
3 eleq1 2114 . . . . . 6 (𝐵 = 𝐴 → (𝐵𝐴𝐴𝐴))
43notbid 600 . . . . 5 (𝐵 = 𝐴 → (¬ 𝐵𝐴 ↔ ¬ 𝐴𝐴))
52, 4syl5ibrcom 150 . . . 4 ( E Fr 𝐴 → (𝐵 = 𝐴 → ¬ 𝐵𝐴))
65necon2ad 2275 . . 3 ( E Fr 𝐴 → (𝐵𝐴𝐵𝐴))
71, 6anim12ii 329 . 2 ((Tr 𝐴 ∧ E Fr 𝐴) → (𝐵𝐴 → (𝐵𝐴𝐵𝐴)))
873impia 1110 1 ((Tr 𝐴 ∧ E Fr 𝐴𝐵𝐴) → (𝐵𝐴𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  w3a 894   = wceq 1257  wcel 1407  wne 2218  wss 2942  Tr wtr 3879   E cep 4049   Fr wfr 4090
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 552  ax-in2 553  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-pow 3952  ax-pr 3969
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ne 2219  df-ral 2326  df-v 2574  df-dif 2945  df-un 2947  df-in 2949  df-ss 2956  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-br 3790  df-opab 3844  df-tr 3880  df-eprel 4051  df-frfor 4093  df-frind 4094
This theorem is referenced by: (None)
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