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Theorem infeq3 6895
Description: Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
Assertion
Ref Expression
infeq3  |-  ( R  =  S  -> inf ( A ,  B ,  R
)  = inf ( A ,  B ,  S
) )

Proof of Theorem infeq3
StepHypRef Expression
1 cnveq 4708 . . 3  |-  ( R  =  S  ->  `' R  =  `' S
)
2 supeq3 6870 . . 3  |-  ( `' R  =  `' S  ->  sup ( A ,  B ,  `' R
)  =  sup ( A ,  B ,  `' S ) )
31, 2syl 14 . 2  |-  ( R  =  S  ->  sup ( A ,  B ,  `' R )  =  sup ( A ,  B ,  `' S ) )
4 df-inf 6865 . 2  |- inf ( A ,  B ,  R
)  =  sup ( A ,  B ,  `' R )
5 df-inf 6865 . 2  |- inf ( A ,  B ,  S
)  =  sup ( A ,  B ,  `' S )
63, 4, 53eqtr4g 2195 1  |-  ( R  =  S  -> inf ( A ,  B ,  R
)  = inf ( A ,  B ,  S
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331   `'ccnv 4533   supcsup 6862  infcinf 6863
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 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-in 3072  df-ss 3079  df-uni 3732  df-br 3925  df-opab 3985  df-cnv 4542  df-sup 6864  df-inf 6865
This theorem is referenced by: (None)
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