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Theorem infeq123d 8971
 Description: Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
Hypotheses
Ref Expression
infeq123d.a (𝜑𝐴 = 𝐷)
infeq123d.b (𝜑𝐵 = 𝐸)
infeq123d.c (𝜑𝐶 = 𝐹)
Assertion
Ref Expression
infeq123d (𝜑 → inf(𝐴, 𝐵, 𝐶) = inf(𝐷, 𝐸, 𝐹))

Proof of Theorem infeq123d
StepHypRef Expression
1 infeq123d.a . . 3 (𝜑𝐴 = 𝐷)
2 infeq123d.b . . 3 (𝜑𝐵 = 𝐸)
3 infeq123d.c . . . 4 (𝜑𝐶 = 𝐹)
43cnveqd 5716 . . 3 (𝜑𝐶 = 𝐹)
51, 2, 4supeq123d 8940 . 2 (𝜑 → sup(𝐴, 𝐵, 𝐶) = sup(𝐷, 𝐸, 𝐹))
6 df-inf 8933 . 2 inf(𝐴, 𝐵, 𝐶) = sup(𝐴, 𝐵, 𝐶)
7 df-inf 8933 . 2 inf(𝐷, 𝐸, 𝐹) = sup(𝐷, 𝐸, 𝐹)
85, 6, 73eqtr4g 2819 1 (𝜑 → inf(𝐴, 𝐵, 𝐶) = inf(𝐷, 𝐸, 𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1539  ◡ccnv 5524  supcsup 8930  infcinf 8931 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-in 3866  df-ss 3876  df-uni 4800  df-br 5034  df-opab 5096  df-cnv 5533  df-sup 8932  df-inf 8933 This theorem is referenced by:  wsuceq123  33355  wlimeq12  33360
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