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Theorem infeq123d 7046
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 4821 . . 3 (𝜑𝐶 = 𝐹)
51, 2, 4supeq123d 7021 . 2 (𝜑 → sup(𝐴, 𝐵, 𝐶) = sup(𝐷, 𝐸, 𝐹))
6 df-inf 7015 . 2 inf(𝐴, 𝐵, 𝐶) = sup(𝐴, 𝐵, 𝐶)
7 df-inf 7015 . 2 inf(𝐷, 𝐸, 𝐹) = sup(𝐷, 𝐸, 𝐹)
85, 6, 73eqtr4g 2247 1 (𝜑 → inf(𝐴, 𝐵, 𝐶) = inf(𝐷, 𝐸, 𝐹))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  ccnv 4643  supcsup 7012  infcinf 7013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-in 3150  df-ss 3157  df-uni 3825  df-br 4019  df-opab 4080  df-cnv 4652  df-sup 7014  df-inf 7015
This theorem is referenced by: (None)
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