Theorem prodeq12si 36184

Description: Equality inference for product. General version of prodeq2si 36183. (Contributed by GG, 1-Sep-2025.)

Hypotheses

Ref	Expression
prodeq12si.1	⊢ 𝐴 = 𝐵
prodeq12si.2	⊢ 𝐶 = 𝐷

Assertion

Ref	Expression
prodeq12si	⊢ ∏𝑥 ∈ 𝐴 𝐶 = ∏𝑥 ∈ 𝐵 𝐷

Proof of Theorem prodeq12si

Step	Hyp	Ref	Expression
1		prodeq12si.1	. . 3 ⊢ 𝐴 = 𝐵
2	1	prodeq1i 15948	. 2 ⊢ ∏𝑥 ∈ 𝐴 𝐶 = ∏𝑥 ∈ 𝐵 𝐶
3		prodeq12si.2	. . 3 ⊢ 𝐶 = 𝐷
4	3	prodeq2si 36183	. 2 ⊢ ∏𝑥 ∈ 𝐵 𝐶 = ∏𝑥 ∈ 𝐵 𝐷
5	2, 4	eqtri 2764	1 ⊢ ∏𝑥 ∈ 𝐴 𝐶 = ∏𝑥 ∈ 𝐵 𝐷

Syntax hints:   = wceq 1540  ∏cprod 15935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-opab 5204  df-mpt 5224  df-xp 5689  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-pred 6319  df-iota 6512  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-ov 7432  df-oprab 7433  df-mpo 7434  df-frecs 8302  df-wrecs 8333  df-recs 8407  df-rdg 8446  df-seq 14039  df-prod 15936

This theorem is referenced by: (None)