![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 3149 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) |
3 | ixpeq2 8458 | . 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 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 Xcixp 8444 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ral 3111 df-v 3443 df-in 3888 df-ss 3898 df-ixp 8445 |
This theorem is referenced by: ixpeq2dv 8460 dfac9 9547 xpsrnbas 16836 funcpropd 17162 natpropd 17238 prdsmgp 19356 frlmip 20467 elptr2 22179 dfac14 22223 xkoptsub 22259 prdsxmslem2 23136 rrxip 23994 ptrest 35056 prdsbnd2 35233 hoidmvlelem3 43236 ovnhoilem1 43240 ovnhoilem2 43241 hoicoto2 43244 ovnlecvr2 43249 ovncvr2 43250 ovnovollem1 43295 ovnovollem2 43296 hoimbl2 43304 vonhoire 43311 iccvonmbllem 43317 vonioolem2 43320 vonicclem2 43323 vonn0ioo2 43329 vonn0icc2 43331 |
Copyright terms: Public domain | W3C validator |