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Theorem prodeq1i 11562
Description: Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.)
Hypothesis
Ref Expression
prodeq1i.1 𝐴 = 𝐵
Assertion
Ref Expression
prodeq1i 𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶
Distinct variable groups:   𝐴,𝑘   𝐵,𝑘
Allowed substitution hint:   𝐶(𝑘)

Proof of Theorem prodeq1i
StepHypRef Expression
1 prodeq1i.1 . 2 𝐴 = 𝐵
2 prodeq1 11554 . 2 (𝐴 = 𝐵 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)
31, 2ax-mp 5 1 𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶
Colors of variables: wff set class
Syntax hints:   = wceq 1353  cprod 11551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-if 3535  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-opab 4064  df-mpt 4065  df-cnv 4633  df-dm 4635  df-rn 4636  df-res 4637  df-iota 5177  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-ov 5875  df-oprab 5876  df-mpo 5877  df-recs 6303  df-frec 6389  df-seqfrec 10441  df-proddc 11552
This theorem is referenced by:  prodeq12i  11564  fprodfac  11616  fprodxp  11625
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