Theorem entr3i 8559

Description: A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)

Hypotheses

Ref	Expression
entr3i.1	⊢ 𝐴 ≈ 𝐵
entr3i.2	⊢ 𝐴 ≈ 𝐶

Assertion

Ref	Expression
entr3i	⊢ 𝐵 ≈ 𝐶

Proof of Theorem entr3i

Step	Hyp	Ref	Expression
1		entr3i.1	. . 3 ⊢ 𝐴 ≈ 𝐵
2	1	ensymi 8553	. 2 ⊢ 𝐵 ≈ 𝐴
3		entr3i.2	. 2 ⊢ 𝐴 ≈ 𝐶
4	2, 3	entri 8557	1 ⊢ 𝐵 ≈ 𝐶

Syntax hints:   class class class wbr 5058   ≈ cen 8500

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-er 8283  df-en 8504

This theorem is referenced by:  cpnnen  15576