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Mirrors > Home > MPE Home > Th. List > ixpeq2dv | Structured version Visualization version GIF version |
Description: Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.) |
Ref | Expression |
---|---|
ixpeq2dv.1 | ⊢ (𝜑 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
ixpeq2dv | ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ixpeq2dv.1 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐶) | |
2 | 1 | adantr 481 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) |
3 | 2 | ixpeq2dva 7923 | 1 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 Xcixp 7908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-in 3581 df-ss 3588 df-ixp 7909 |
This theorem is referenced by: prdsval 16115 brssc 16474 isfunc 16524 natfval 16606 isnat 16607 dprdval 18402 elpt 21375 elptr 21376 dfac14 21421 hoicvrrex 40770 ovncvrrp 40778 ovnsubaddlem1 40784 ovnsubadd 40786 hoidmvlelem3 40811 hoidmvle 40814 ovnhoilem1 40815 ovnhoilem2 40816 ovnhoi 40817 hspval 40823 ovncvr2 40825 hspmbllem2 40841 hspmbl 40843 hoimbl 40845 opnvonmbl 40848 ovnovollem1 40870 ovnovollem3 40872 iinhoiicclem 40887 iinhoiicc 40888 vonioolem2 40895 vonioo 40896 vonicclem2 40898 vonicc 40899 |
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