Theorem infeq1 9235

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 9204	. 2 ⊢ (𝐵 = 𝐶 → sup(𝐵, 𝐴, ◡𝑅) = sup(𝐶, 𝐴, ◡𝑅))
2		df-inf 9202	. 2 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅)
3		df-inf 9202	. 2 ⊢ inf(𝐶, 𝐴, 𝑅) = sup(𝐶, 𝐴, ◡𝑅)
4	1, 2, 3	3eqtr4g 2803	1 ⊢ (𝐵 = 𝐶 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))

Syntax hints:   → wi 4   = wceq 1539  ^◡ccnv 5588  supcsup 9199  infcinf 9200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-in 3894  df-ss 3904  df-uni 4840  df-sup 9201  df-inf 9202

This theorem is referenced by:  infeq1d  9236  infeq1i  9237  ramcl2lem  16710  odfval  19140  odval  19142  submod  19174  ioorval  24738  uniioombllem6  24752  infleinf  42911  infxrpnf  42986  prproropf1olem2  44956  prproropf1olem3  44957  prproropf1olem4  44958  prproropf1o  44959  prproropreud  44961