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| Mirrors > Home > MPE Home > Th. List > Mathboxes > ixpeq2d | Structured version Visualization version GIF version | ||
| Description: Equality theorem for infinite Cartesian product. (Contributed by Glauco Siliprandi, 11-Oct-2020.) | 
| Ref | Expression | 
|---|---|
| ixpeq2d.1 | ⊢ Ⅎ𝑥𝜑 | 
| ixpeq2d.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) | 
| Ref | Expression | 
|---|---|
| ixpeq2d | ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ixpeq2d.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
| 2 | ixpeq2d.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) | |
| 3 | 2 | ex 412 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 = 𝐶)) | 
| 4 | 1, 3 | ralrimi 3257 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) | 
| 5 | ixpeq2 8951 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶) | |
| 6 | 4, 5 | syl 17 | 1 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ 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-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 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: hoicvrrex 46571 ovnlecvr 46573 ovnhoilem1 46616 hoi2toco 46622 ovnlecvr2 46625 opnvonmbllem1 46647 | 
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