Theorem infeq1 9380

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 9348	. 2 ⊢ (𝐵 = 𝐶 → sup(𝐵, 𝐴, ◡𝑅) = sup(𝐶, 𝐴, ◡𝑅))
2		df-inf 9346	. 2 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅)
3		df-inf 9346	. 2 ⊢ inf(𝐶, 𝐴, 𝑅) = sup(𝐶, 𝐴, ◡𝑅)
4	1, 2, 3	3eqtr4g 2796	1 ⊢ (𝐵 = 𝐶 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))

Syntax hints:   → wi 4   = wceq 1541  ^◡ccnv 5623  supcsup 9343  infcinf 9344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-ss 3918  df-uni 4864  df-sup 9345  df-inf 9346

This theorem is referenced by:  infeq1d  9381  infeq1i  9382  ramcl2lem  16937  odfval  19461  odval  19463  submod  19498  ioorval  25531  uniioombllem6  25545  infleinf  45616  infxrpnf  45690  prproropf1olem2  47750  prproropf1olem3  47751  prproropf1olem4  47752  prproropf1o  47753  prproropreud  47755