Theorem prodeq12si 36152

Description: Equality inference for product. General version of prodeq2si 36151. (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 15921	. 2 ⊢ ∏𝑥 ∈ 𝐴 𝐶 = ∏𝑥 ∈ 𝐵 𝐶
3		prodeq12si.2	. . 3 ⊢ 𝐶 = 𝐷
4	3	prodeq2si 36151	. 2 ⊢ ∏𝑥 ∈ 𝐵 𝐶 = ∏𝑥 ∈ 𝐵 𝐷
5	2, 4	eqtri 2757	1 ⊢ ∏𝑥 ∈ 𝐴 𝐶 = ∏𝑥 ∈ 𝐵 𝐷

Syntax hints:   = wceq 1539  ∏cprod 15908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4882  df-br 5118  df-opab 5180  df-mpt 5200  df-xp 5658  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6288  df-iota 6481  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-ov 7403  df-oprab 7404  df-mpo 7405  df-frecs 8275  df-wrecs 8306  df-recs 8380  df-rdg 8419  df-seq 14010  df-prod 15909

This theorem is referenced by: (None)