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Theorem prodeq1d 11394
 Description: Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.)
Hypothesis
Ref Expression
prodeq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
prodeq1d (𝜑 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)
Distinct variable groups:   𝐴,𝑘   𝐵,𝑘
Allowed substitution hints:   𝜑(𝑘)   𝐶(𝑘)

Proof of Theorem prodeq1d
StepHypRef Expression
1 prodeq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 prodeq1 11383 . 2 (𝐴 = 𝐵 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)
31, 2syl 14 1 (𝜑 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1332  ∏cprod 11380 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2123 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1732  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-ral 2423  df-rex 2424  df-v 2693  df-un 3082  df-in 3084  df-ss 3091  df-if 3482  df-sn 3540  df-pr 3541  df-op 3543  df-uni 3747  df-br 3940  df-opab 4000  df-mpt 4001  df-cnv 4559  df-dm 4561  df-rn 4562  df-res 4563  df-iota 5100  df-f 5139  df-f1 5140  df-fo 5141  df-f1o 5142  df-fv 5143  df-ov 5789  df-oprab 5790  df-mpo 5791  df-recs 6214  df-frec 6300  df-seqfrec 10279  df-proddc 11381 This theorem is referenced by:  prodeq12dv  11399  prodeq12rdv  11400  fprodf1o  11418
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