Theorem infeq1 6891

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 6866	. 2 ⊢ (𝐵 = 𝐶 → sup(𝐵, 𝐴, ◡𝑅) = sup(𝐶, 𝐴, ◡𝑅))
2		df-inf 6865	. 2 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅)
3		df-inf 6865	. 2 ⊢ inf(𝐶, 𝐴, 𝑅) = sup(𝐶, 𝐴, ◡𝑅)
4	1, 2, 3	3eqtr4g 2195	1 ⊢ (𝐵 = 𝐶 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))

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-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-uni 3732  df-sup 6864  df-inf 6865

This theorem is referenced by:  infeq1d  6892  infeq1i  6893