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Theorem infeq3 6654
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 4578 . . 3  |-  ( R  =  S  ->  `' R  =  `' S
)
2 supeq3 6629 . . 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 6624 . 2  |- inf ( A ,  B ,  R
)  =  sup ( A ,  B ,  `' R )
5 df-inf 6624 . 2  |- inf ( A ,  B ,  S
)  =  sup ( A ,  B ,  `' S )
63, 4, 53eqtr4g 2142 1  |-  ( R  =  S  -> inf ( A ,  B ,  R
)  = inf ( A ,  B ,  S
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1287   `'ccnv 4410   supcsup 6621  infcinf 6622
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-rab 2364  df-in 2994  df-ss 3001  df-uni 3637  df-br 3821  df-opab 3875  df-cnv 4419  df-sup 6623  df-inf 6624
This theorem is referenced by: (None)
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