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Theorem tz7.2 5672
Description: Similar to Theorem 7.2 of [TakeutiZaring] p. 35, except that the Axiom of Regularity is not required due to the 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 5276 . . 3 (Tr 𝐴 → (𝐵𝐴𝐵𝐴))
2 efrirr 5669 . . . . 5 ( E Fr 𝐴 → ¬ 𝐴𝐴)
3 eleq1 2827 . . . . . 6 (𝐵 = 𝐴 → (𝐵𝐴𝐴𝐴))
43notbid 318 . . . . 5 (𝐵 = 𝐴 → (¬ 𝐵𝐴 ↔ ¬ 𝐴𝐴))
52, 4syl5ibrcom 247 . . . 4 ( E Fr 𝐴 → (𝐵 = 𝐴 → ¬ 𝐵𝐴))
65necon2ad 2953 . . 3 ( E Fr 𝐴 → (𝐵𝐴𝐵𝐴))
71, 6anim12ii 618 . 2 ((Tr 𝐴 ∧ E Fr 𝐴) → (𝐵𝐴 → (𝐵𝐴𝐵𝐴)))
873impia 1116 1 ((Tr 𝐴 ∧ E Fr 𝐴𝐵𝐴) → (𝐵𝐴𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wss 3963  Tr wtr 5265   E cep 5588   Fr wfr 5638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-fr 5641
This theorem is referenced by:  tz7.7  6412  trelpss  44451
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