Theorem infeq3 9441

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 5860	. . 3 ⊢ (𝑅 = 𝑆 → ◡𝑅 = ◡𝑆)
2		supeq3 9409	. . 3 ⊢ (◡𝑅 = ◡𝑆 → sup(𝐴, 𝐵, ◡𝑅) = sup(𝐴, 𝐵, ◡𝑆))
3	1, 2	syl 18	. 2 ⊢ (𝑅 = 𝑆 → sup(𝐴, 𝐵, ◡𝑅) = sup(𝐴, 𝐵, ◡𝑆))
4		df-inf 9403	. 2 ⊢ inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)
5		df-inf 9403	. 2 ⊢ inf(𝐴, 𝐵, 𝑆) = sup(𝐴, 𝐵, ◡𝑆)
6	3, 4, 5	3eqtr4g 2829	1 ⊢ (𝑅 = 𝑆 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐵, 𝑆))

Syntax hints:   → wi 4   = wceq 1567  ^◡ccnv 5661  supcsup 9400  infcinf 9401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3423  df-v 3463  df-ss 3928  df-uni 4875  df-br 5112  df-opab 5176  df-cnv 5670  df-sup 9402  df-inf 9403

This theorem is referenced by: (None)