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| Mirrors > Home > MPE Home > Th. List > ixpeq2dva | Structured version Visualization version GIF version | ||
| Description: Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.) |
| Ref | Expression |
|---|---|
| ixpeq2dva.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) |
| Ref | Expression |
|---|---|
| ixpeq2dva | ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ixpeq2dva.1 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) | |
| 2 | 1 | ralrimiva 3146 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) |
| 3 | ixpeq2 8951 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶) | |
| 4 | 2, 3 | syl 17 | 1 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Xcixp 8937 |
| 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 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-ss 3968 df-ixp 8938 |
| This theorem is referenced by: ixpeq2dv 8953 dfac9 10177 xpsrnbas 17616 funcpropd 17947 natpropd 18024 prdsmgp 20148 frlmip 21798 elptr2 23582 dfac14 23626 xkoptsub 23662 prdsxmslem2 24542 rrxip 25424 ptrest 37626 prdsbnd2 37802 hoidmvlelem3 46612 ovnhoilem1 46616 ovnhoilem2 46617 hoicoto2 46620 ovnlecvr2 46625 ovncvr2 46626 ovnovollem1 46671 ovnovollem2 46672 hoimbl2 46680 vonhoire 46687 iccvonmbllem 46693 vonioolem2 46696 vonicclem2 46699 vonn0ioo2 46705 vonn0icc2 46707 |
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