Theorem infeq123d 9455

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 5859	. . 3 ⊢ (𝜑 → ◡𝐶 = ◡𝐹)
5	1, 2, 4	supeq123d 9423	. 2 ⊢ (𝜑 → sup(𝐴, 𝐵, ◡𝐶) = sup(𝐷, 𝐸, ◡𝐹))
6		df-inf 9416	. 2 ⊢ inf(𝐴, 𝐵, 𝐶) = sup(𝐴, 𝐵, ◡𝐶)
7		df-inf 9416	. 2 ⊢ inf(𝐷, 𝐸, 𝐹) = sup(𝐷, 𝐸, ◡𝐹)
8	5, 6, 7	3eqtr4g 2822	1 ⊢ (𝜑 → inf(𝐴, 𝐵, 𝐶) = inf(𝐷, 𝐸, 𝐹))

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ^◡ccnv 5658  supcsup 9413  infcinf 9414

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-ss 3919  df-uni 4871  df-br 5108  df-opab 5172  df-cnv 5667  df-sup 9415  df-inf 9416

This theorem is used by:  wsuceq123  36376  wlimeq12  36381