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Theorem enen1 8051
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 7956 . . 3 (𝐴𝐵𝐵𝐴)
2 entr 7959 . . 3 ((𝐵𝐴𝐴𝐶) → 𝐵𝐶)
31, 2sylan 488 . 2 ((𝐴𝐵𝐴𝐶) → 𝐵𝐶)
4 entr 7959 . 2 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
53, 4impbida 876 1 (𝐴𝐵 → (𝐴𝐶𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   class class class wbr 4618  cen 7903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-er 7694  df-en 7907
This theorem is referenced by:  onomeneq  8101  enfi  8127  alephexp2  9354  pmtrfmvdn0  17810
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