Theorem infeq123d 9361

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

Hypotheses

Ref	Expression
infeq123d.a	⊢ (𝜑 → 𝐴 = 𝐷)
infeq123d.b	⊢ (𝜑 → 𝐵 = 𝐸)
infeq123d.c	⊢ (𝜑 → 𝐶 = 𝐹)

Assertion

Ref	Expression
infeq123d	⊢ (𝜑 → inf(𝐴, 𝐵, 𝐶) = inf(𝐷, 𝐸, 𝐹))

Proof of Theorem infeq123d

Step	Hyp	Ref	Expression
1		infeq123d.a	. . 3 ⊢ (𝜑 → 𝐴 = 𝐷)
2		infeq123d.b	. . 3 ⊢ (𝜑 → 𝐵 = 𝐸)
3		infeq123d.c	. . . 4 ⊢ (𝜑 → 𝐶 = 𝐹)
4	3	cnveqd 5810	. . 3 ⊢ (𝜑 → ◡𝐶 = ◡𝐹)
5	1, 2, 4	supeq123d 9329	. 2 ⊢ (𝜑 → sup(𝐴, 𝐵, ◡𝐶) = sup(𝐷, 𝐸, ◡𝐹))
6		df-inf 9322	. 2 ⊢ inf(𝐴, 𝐵, 𝐶) = sup(𝐴, 𝐵, ◡𝐶)
7		df-inf 9322	. 2 ⊢ inf(𝐷, 𝐸, 𝐹) = sup(𝐷, 𝐸, ◡𝐹)
8	5, 6, 7	3eqtr4g 2791	1 ⊢ (𝜑 → inf(𝐴, 𝐵, 𝐶) = inf(𝐷, 𝐸, 𝐹))

Syntax hints:   → wi 4   = wceq 1541  ^◡ccnv 5610  supcsup 9319  infcinf 9320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-ss 3914  df-uni 4855  df-br 5087  df-opab 5149  df-cnv 5619  df-sup 9321  df-inf 9322

This theorem is referenced by:  wsuceq123  35848  wlimeq12  35853