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Theorem enen1 8141
Description: Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
Assertion
Ref Expression
enen1 (𝐴𝐵 → (𝐴𝐶𝐵𝐶))

Proof of Theorem enen1
StepHypRef Expression
1 ensym 8046 . . 3 (𝐴𝐵𝐵𝐴)
2 entr 8049 . . 3 ((𝐵𝐴𝐴𝐶) → 𝐵𝐶)
31, 2sylan 487 . 2 ((𝐴𝐵𝐴𝐶) → 𝐵𝐶)
4 entr 8049 . 2 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
53, 4impbida 895 1 (𝐴𝐵 → (𝐴𝐶𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   class class class wbr 4685  cen 7994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-er 7787  df-en 7998
This theorem is referenced by:  onomeneq  8191  enfi  8217  alephexp2  9441  pmtrfmvdn0  17928
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