Theorem infeq1 9384

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 9352	. 2 ⊢ (𝐵 = 𝐶 → sup(𝐵, 𝐴, ◡𝑅) = sup(𝐶, 𝐴, ◡𝑅))
2		df-inf 9350	. 2 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅)
3		df-inf 9350	. 2 ⊢ inf(𝐶, 𝐴, 𝑅) = sup(𝐶, 𝐴, ◡𝑅)
4	1, 2, 3	3eqtr4g 2797	1 ⊢ (𝐵 = 𝐶 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))

Syntax hints:   → wi 4   = wceq 1542  ^◡ccnv 5624  supcsup 9347  infcinf 9348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-ss 3907  df-uni 4852  df-sup 9349  df-inf 9350

This theorem is referenced by:  infeq1d  9385  infeq1i  9386  ramcl2lem  16974  odfval  19501  odval  19503  submod  19538  ioorval  25554  uniioombllem6  25568  infleinf  45822  infxrpnf  45895  prproropf1olem2  47979  prproropf1olem3  47980  prproropf1olem4  47981  prproropf1o  47982  prproropreud  47984