Theorem infeq123d 8675

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 5543	. . 3 ⊢ (𝜑 → ◡𝐶 = ◡𝐹)
5	1, 2, 4	supeq123d 8644	. 2 ⊢ (𝜑 → sup(𝐴, 𝐵, ◡𝐶) = sup(𝐷, 𝐸, ◡𝐹))
6		df-inf 8637	. 2 ⊢ inf(𝐴, 𝐵, 𝐶) = sup(𝐴, 𝐵, ◡𝐶)
7		df-inf 8637	. 2 ⊢ inf(𝐷, 𝐸, 𝐹) = sup(𝐷, 𝐸, ◡𝐹)
8	5, 6, 7	3eqtr4g 2838	1 ⊢ (𝜑 → inf(𝐴, 𝐵, 𝐶) = inf(𝐷, 𝐸, 𝐹))

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 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-ext 2753

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 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-in 3798  df-ss 3805  df-uni 4672  df-br 4887  df-opab 4949  df-cnv 5363  df-sup 8636  df-inf 8637

This theorem is referenced by:  wsuceq123  32362  wlimeq12  32367