ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  infeq123d Unicode version

Theorem infeq123d 6655
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 4580 . . 3  |-  ( ph  ->  `' C  =  `' F )
51, 2, 4supeq123d 6630 . 2  |-  ( ph  ->  sup ( A ,  B ,  `' C
)  =  sup ( D ,  E ,  `' F ) )
6 df-inf 6624 . 2  |- inf ( A ,  B ,  C
)  =  sup ( A ,  B ,  `' C )
7 df-inf 6624 . 2  |- inf ( D ,  E ,  F
)  =  sup ( D ,  E ,  `' F )
85, 6, 73eqtr4g 2142 1  |-  ( ph  -> inf ( A ,  B ,  C )  = inf ( D ,  E ,  F ) )
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)
  Copyright terms: Public domain W3C validator