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| Mirrors > Home > MPE Home > Th. List > ixpeq1d | Structured version Visualization version GIF version | ||
| Description: Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.) |
| Ref | Expression |
|---|---|
| ixpeq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
| Ref | Expression |
|---|---|
| ixpeq1d | ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ixpeq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 2 | ixpeq1 8948 | . 2 ⊢ (𝐴 = 𝐵 → X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 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-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-fn 6564 df-ixp 8938 |
| This theorem is referenced by: elixpsn 8977 ixpsnf1o 8978 dfac9 10177 prdsval 17500 isfunc 17909 funcpropd 17947 natfval 17994 natpropd 18024 dprdval 20023 ptval 23578 dfac14 23626 ptuncnv 23815 ptunhmeo 23816 hoidmvle 46615 hoimbl 46646 |
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