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Theorem tz7.2 4375
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 4128 . . 3 (Tr 𝐴 → (𝐵𝐴𝐵𝐴))
2 efrirr 4374 . . . . 5 ( E Fr 𝐴 → ¬ 𝐴𝐴)
3 eleq1 2252 . . . . . 6 (𝐵 = 𝐴 → (𝐵𝐴𝐴𝐴))
43notbid 668 . . . . 5 (𝐵 = 𝐴 → (¬ 𝐵𝐴 ↔ ¬ 𝐴𝐴))
52, 4syl5ibrcom 157 . . . 4 ( E Fr 𝐴 → (𝐵 = 𝐴 → ¬ 𝐵𝐴))
65necon2ad 2417 . . 3 ( E Fr 𝐴 → (𝐵𝐴𝐵𝐴))
71, 6anim12ii 343 . 2 ((Tr 𝐴 ∧ E Fr 𝐴) → (𝐵𝐴 → (𝐵𝐴𝐵𝐴)))
873impia 1202 1 ((Tr 𝐴 ∧ E Fr 𝐴𝐵𝐴) → (𝐵𝐴𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  w3a 980   = wceq 1364  wcel 2160  wne 2360  wss 3144  Tr wtr 4119   E cep 4308   Fr wfr 4349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-br 4022  df-opab 4083  df-tr 4120  df-eprel 4310  df-frfor 4352  df-frind 4353
This theorem is referenced by: (None)
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