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Theorem tz7.2 5621
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 5237 . . 3 (Tr 𝐴 → (𝐵𝐴𝐵𝐴))
2 efrirr 5618 . . . . 5 ( E Fr 𝐴 → ¬ 𝐴𝐴)
3 eleq1 2822 . . . . . 6 (𝐵 = 𝐴 → (𝐵𝐴𝐴𝐴))
43notbid 318 . . . . 5 (𝐵 = 𝐴 → (¬ 𝐵𝐴 ↔ ¬ 𝐴𝐴))
52, 4syl5ibrcom 247 . . . 4 ( E Fr 𝐴 → (𝐵 = 𝐴 → ¬ 𝐵𝐴))
65necon2ad 2955 . . 3 ( E Fr 𝐴 → (𝐵𝐴𝐵𝐴))
71, 6anim12ii 619 . 2 ((Tr 𝐴 ∧ E Fr 𝐴) → (𝐵𝐴 → (𝐵𝐴𝐵𝐴)))
873impia 1118 1 ((Tr 𝐴 ∧ E Fr 𝐴𝐵𝐴) → (𝐵𝐴𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wne 2940  wss 3914  Tr wtr 5226   E cep 5540   Fr wfr 5589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-tr 5227  df-eprel 5541  df-fr 5592
This theorem is referenced by:  tz7.7  6347  trelpss  42827
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