Theorem prodeq1i 12311

Description: Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.)

Hypothesis

Ref	Expression
prodeq1i.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
prodeq1i	⊢ ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶

Distinct variable groups:   𝐴,𝑘   𝐵,𝑘

Allowed substitution hint:   𝐶(𝑘)

Proof of Theorem prodeq1i

Step	Hyp	Ref	Expression
1		prodeq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		prodeq1 12303	. 2 ⊢ (𝐴 = 𝐵 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶)
3	1, 2	ax-mp 5	1 ⊢ ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶

Syntax hints:   = wceq 1402  ∏cprod 12300

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-mpt 4192  df-cnv 4780  df-dm 4782  df-rn 4783  df-res 4784  df-iota 5335  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-recs 6570  df-frec 6656  df-seqfrec 10868  df-proddc 12301

This theorem is referenced by:  prodeq12i  12313  fprodfac  12365  fprodxp  12374