Theorem infeq3 9518

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

Assertion

Ref	Expression
infeq3	⊢ (𝑅 = 𝑆 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐵, 𝑆))

Proof of Theorem infeq3

Step	Hyp	Ref	Expression
1		cnveq 5887	. . 3 ⊢ (𝑅 = 𝑆 → ◡𝑅 = ◡𝑆)
2		supeq3 9487	. . 3 ⊢ (◡𝑅 = ◡𝑆 → sup(𝐴, 𝐵, ◡𝑅) = sup(𝐴, 𝐵, ◡𝑆))
3	1, 2	syl 17	. 2 ⊢ (𝑅 = 𝑆 → sup(𝐴, 𝐵, ◡𝑅) = sup(𝐴, 𝐵, ◡𝑆))
4		df-inf 9481	. 2 ⊢ inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)
5		df-inf 9481	. 2 ⊢ inf(𝐴, 𝐵, 𝑆) = sup(𝐴, 𝐵, ◡𝑆)
6	3, 4, 5	3eqtr4g 2800	1 ⊢ (𝑅 = 𝑆 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐵, 𝑆))

Syntax hints:   → wi 4   = wceq 1537  ^◡ccnv 5688  supcsup 9478  infcinf 9479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-ss 3980  df-uni 4913  df-br 5149  df-opab 5211  df-cnv 5697  df-sup 9480  df-inf 9481

This theorem is referenced by: (None)