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Theorem infeq123d 9471
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 5865 . . 3 (𝜑𝐶 = 𝐹)
51, 2, 4supeq123d 9440 . 2 (𝜑 → sup(𝐴, 𝐵, 𝐶) = sup(𝐷, 𝐸, 𝐹))
6 df-inf 9433 . 2 inf(𝐴, 𝐵, 𝐶) = sup(𝐴, 𝐵, 𝐶)
7 df-inf 9433 . 2 inf(𝐷, 𝐸, 𝐹) = sup(𝐷, 𝐸, 𝐹)
85, 6, 73eqtr4g 2789 1 (𝜑 → inf(𝐴, 𝐵, 𝐶) = inf(𝐷, 𝐸, 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  ccnv 5665  supcsup 9430  infcinf 9431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-in 3947  df-ss 3957  df-uni 4900  df-br 5139  df-opab 5201  df-cnv 5674  df-sup 9432  df-inf 9433
This theorem is referenced by:  wsuceq123  35247  wlimeq12  35252
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