Theorem prodeq12si 36147

Description: Equality inference for product. General version of prodeq2si 36146. (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 15938	. 2 ⊢ ∏𝑥 ∈ 𝐴 𝐶 = ∏𝑥 ∈ 𝐵 𝐶
3		prodeq12si.2	. . 3 ⊢ 𝐶 = 𝐷
4	3	prodeq2si 36146	. 2 ⊢ ∏𝑥 ∈ 𝐵 𝐶 = ∏𝑥 ∈ 𝐵 𝐷
5	2, 4	eqtri 2761	1 ⊢ ∏𝑥 ∈ 𝐴 𝐶 = ∏𝑥 ∈ 𝐵 𝐷

Syntax hints:   = wceq 1535  ∏cprod 15925

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-br 5150  df-opab 5212  df-mpt 5233  df-xp 5689  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-pred 6317  df-iota 6510  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7428  df-oprab 7429  df-mpo 7430  df-frecs 8299  df-wrecs 8330  df-recs 8404  df-rdg 8443  df-seq 14029  df-prod 15926

This theorem is referenced by: (None)