Theorem prodeq12si 36565

Description: Equality inference for product. General version of prodeq2si 36564. (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 15946	. 2 ⊢ ∏𝑥 ∈ 𝐴 𝐶 = ∏𝑥 ∈ 𝐵 𝐶
3		prodeq12si.2	. . 3 ⊢ 𝐶 = 𝐷
4	3	prodeq2si 36564	. 2 ⊢ ∏𝑥 ∈ 𝐵 𝐶 = ∏𝑥 ∈ 𝐵 𝐷
5	2, 4	eqtri 2785	1 ⊢ ∏𝑥 ∈ 𝐴 𝐶 = ∏𝑥 ∈ 𝐵 𝐷

Syntax hints:   = wceq 1560  ∏cprod 15933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-xp 5653  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-iota 6477  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-seq 14015  df-prod 15934

This theorem is referenced by: (None)