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| Description: A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.) | 
| Ref | Expression | 
|---|---|
| entri.1 | ⊢ 𝐴 ≈ 𝐵 | 
| entri.2 | ⊢ 𝐵 ≈ 𝐶 | 
| Ref | Expression | 
|---|---|
| entri | ⊢ 𝐴 ≈ 𝐶 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | entri.1 | . 2 ⊢ 𝐴 ≈ 𝐵 | |
| 2 | entri.2 | . 2 ⊢ 𝐵 ≈ 𝐶 | |
| 3 | entr 9046 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶) | |
| 4 | 1, 2, 3 | mp2an 692 | 1 ⊢ 𝐴 ≈ 𝐶 | 
| Colors of variables: wff setvar class | 
| Syntax hints: class class class wbr 5143 ≈ cen 8982 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-er 8745 df-en 8986 | 
| This theorem is referenced by: entr2i 9049 entr3i 9050 entr4i 9051 infxpenc2 10062 cfpwsdom 10624 hashxplem 14472 xpnnen 16247 qnnen 16249 rpnnen 16263 rexpen 16264 odhash 19592 cygctb 19910 met2ndci 24535 re2ndc 24822 iscmet3 25327 dyadmbl 25635 opnmblALT 25638 mbfimaopnlem 25690 aannenlem3 26372 mblfinlem1 37664 heiborlem3 37820 heibor 37828 irrapx1 42839 zenom 45057 qenom 45372 | 
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