Theorem infeq1 6988

Description: Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)

Assertion

Ref	Expression
infeq1	|- ( B = C -> inf ( B , A , R ) = inf ( C , A , R ) )

Proof of Theorem infeq1

Step	Hyp	Ref	Expression
1		supeq1 6963	. 2 |- ( B = C -> sup ( B , A , `' R ) = sup ( C , A , `' R ) )
2		df-inf 6962	. 2 |- inf ( B , A , R ) = sup ( B , A , `' R )
3		df-inf 6962	. 2 |- inf ( C , A , R ) = sup ( C , A , `' R )
4	1, 2, 3	3eqtr4g 2228	1 |- ( B = C -> inf ( B , A , R ) = inf ( C , A , R ) )

Syntax hints:   -> wi 4   = wceq 1348   `'ccnv 4610   supcsup 6959  infcinf 6960

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-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-uni 3797  df-sup 6961  df-inf 6962

This theorem is referenced by:  infeq1d  6989  infeq1i  6990