Theorem infeq2 8673

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

Assertion

Ref	Expression
infeq2	⊢ (𝐵 = 𝐶 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐶, 𝑅))

Proof of Theorem infeq2

Step	Hyp	Ref	Expression
1		supeq2 8642	. 2 ⊢ (𝐵 = 𝐶 → sup(𝐴, 𝐵, ◡𝑅) = sup(𝐴, 𝐶, ◡𝑅))
2		df-inf 8637	. 2 ⊢ inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)
3		df-inf 8637	. 2 ⊢ inf(𝐴, 𝐶, 𝑅) = sup(𝐴, 𝐶, ◡𝑅)
4	1, 2, 3	3eqtr4g 2839	1 ⊢ (𝐵 = 𝐶 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐶, 𝑅))

Syntax hints:   → wi 4   = wceq 1601  ^◡ccnv 5354  supcsup 8634  infcinf 8635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-uni 4672  df-sup 8636  df-inf 8637

This theorem is referenced by: (None)