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Theorem infeq123d 6765
 Description: Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
Hypotheses
Ref Expression
infeq123d.a (𝜑𝐴 = 𝐷)
infeq123d.b (𝜑𝐵 = 𝐸)
infeq123d.c (𝜑𝐶 = 𝐹)
Assertion
Ref Expression
infeq123d (𝜑 → inf(𝐴, 𝐵, 𝐶) = inf(𝐷, 𝐸, 𝐹))

Proof of Theorem infeq123d
StepHypRef Expression
1 infeq123d.a . . 3 (𝜑𝐴 = 𝐷)
2 infeq123d.b . . 3 (𝜑𝐵 = 𝐸)
3 infeq123d.c . . . 4 (𝜑𝐶 = 𝐹)
43cnveqd 4625 . . 3 (𝜑𝐶 = 𝐹)
51, 2, 4supeq123d 6740 . 2 (𝜑 → sup(𝐴, 𝐵, 𝐶) = sup(𝐷, 𝐸, 𝐹))
6 df-inf 6734 . 2 inf(𝐴, 𝐵, 𝐶) = sup(𝐴, 𝐵, 𝐶)
7 df-inf 6734 . 2 inf(𝐷, 𝐸, 𝐹) = sup(𝐷, 𝐸, 𝐹)
85, 6, 73eqtr4g 2146 1 (𝜑 → inf(𝐴, 𝐵, 𝐶) = inf(𝐷, 𝐸, 𝐹))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1290  ◡ccnv 4451  supcsup 6731  infcinf 6732 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071 This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-rab 2369  df-in 3006  df-ss 3013  df-uni 3660  df-br 3852  df-opab 3906  df-cnv 4460  df-sup 6733  df-inf 6734 This theorem is referenced by: (None)
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