Theorem infeq3 6522

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 4557	. . 3 ⊢ (𝑅 = 𝑆 → ◡𝑅 = ◡𝑆)
2		supeq3 6497	. . 3 ⊢ (◡𝑅 = ◡𝑆 → sup(𝐴, 𝐵, ◡𝑅) = sup(𝐴, 𝐵, ◡𝑆))
3	1, 2	syl 14	. 2 ⊢ (𝑅 = 𝑆 → sup(𝐴, 𝐵, ◡𝑅) = sup(𝐴, 𝐵, ◡𝑆))
4		df-inf 6492	. 2 ⊢ inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)
5		df-inf 6492	. 2 ⊢ inf(𝐴, 𝐵, 𝑆) = sup(𝐴, 𝐵, ◡𝑆)
6	3, 4, 5	3eqtr4g 2140	1 ⊢ (𝑅 = 𝑆 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐵, 𝑆))

Syntax hints:   → wi 4   = wceq 1285  ^◡ccnv 4390  supcsup 6489  infcinf 6490

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-in 2988  df-ss 2995  df-uni 3622  df-br 3806  df-opab 3860  df-cnv 4399  df-sup 6491  df-inf 6492

This theorem is referenced by: (None)