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Theorem prodeq1iOLD 15949
Description: Obsolete version of prodeq1i 15948 as of 1-Sep-2025. (Contributed by Scott Fenton, 4-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
prodeq1iOLD.1 𝐴 = 𝐵
Assertion
Ref Expression
prodeq1iOLD 𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶
Distinct variable groups:   𝐴,𝑘   𝐵,𝑘
Allowed substitution hint:   𝐶(𝑘)

Proof of Theorem prodeq1iOLD
StepHypRef Expression
1 prodeq1iOLD.1 . 2 𝐴 = 𝐵
2 prodeq1 15939 . 2 (𝐴 = 𝐵 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)
31, 2ax-mp 5 1 𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶
Colors of variables: wff setvar class
Syntax hints:   = wceq 1536  cprod 15935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-xp 5694  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-iota 6515  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-seq 14039  df-prod 15936
This theorem is referenced by: (None)
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