Theorem prodeq12si 36188

Description: Equality inference for product. General version of prodeq2si 36187. (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 15888	. 2 ⊢ ∏𝑥 ∈ 𝐴 𝐶 = ∏𝑥 ∈ 𝐵 𝐶
3		prodeq12si.2	. . 3 ⊢ 𝐶 = 𝐷
4	3	prodeq2si 36187	. 2 ⊢ ∏𝑥 ∈ 𝐵 𝐶 = ∏𝑥 ∈ 𝐵 𝐷
5	2, 4	eqtri 2753	1 ⊢ ∏𝑥 ∈ 𝐴 𝐶 = ∏𝑥 ∈ 𝐵 𝐷

Syntax hints:   = wceq 1540  ∏cprod 15875

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-mpt 5191  df-xp 5646  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6276  df-iota 6466  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-ov 7392  df-oprab 7393  df-mpo 7394  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8380  df-seq 13973  df-prod 15876

This theorem is referenced by: (None)