Theorem prodeq1i 15364

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 15355	. 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 1542  ∏cprod 15351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-un 3848  df-in 3850  df-ss 3860  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-xp 5531  df-cnv 5533  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-iota 6297  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-seq 13461  df-prod 15352

This theorem is referenced by:  prodeq12i  15366  fprodxp  15428  risefac0  15473  fallfacfwd  15482  prmo0  16472  breprexp  32183  etransclem31  43348  etransclem35  43352  hoidmv1le  43674  fmtnorec2  44529