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Theorem infeq123d 6903
Description: Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
Hypotheses
Ref Expression
infeq123d.a  |-  ( ph  ->  A  =  D )
infeq123d.b  |-  ( ph  ->  B  =  E )
infeq123d.c  |-  ( ph  ->  C  =  F )
Assertion
Ref Expression
infeq123d  |-  ( ph  -> inf ( A ,  B ,  C )  = inf ( D ,  E ,  F ) )

Proof of Theorem infeq123d
StepHypRef Expression
1 infeq123d.a . . 3  |-  ( ph  ->  A  =  D )
2 infeq123d.b . . 3  |-  ( ph  ->  B  =  E )
3 infeq123d.c . . . 4  |-  ( ph  ->  C  =  F )
43cnveqd 4715 . . 3  |-  ( ph  ->  `' C  =  `' F )
51, 2, 4supeq123d 6878 . 2  |-  ( ph  ->  sup ( A ,  B ,  `' C
)  =  sup ( D ,  E ,  `' F ) )
6 df-inf 6872 . 2  |- inf ( A ,  B ,  C
)  =  sup ( A ,  B ,  `' C )
7 df-inf 6872 . 2  |- inf ( D ,  E ,  F
)  =  sup ( D ,  E ,  `' F )
85, 6, 73eqtr4g 2197 1  |-  ( ph  -> inf ( A ,  B ,  C )  = inf ( D ,  E ,  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331   `'ccnv 4538   supcsup 6869  infcinf 6870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-in 3077  df-ss 3084  df-uni 3737  df-br 3930  df-opab 3990  df-cnv 4547  df-sup 6871  df-inf 6872
This theorem is referenced by: (None)
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