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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 8567 | . 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 1543 Xcixp 8556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-fn 6361 df-ixp 8557 |
This theorem is referenced by: elixpsn 8596 ixpsnf1o 8597 dfac9 9715 prdsval 16914 isfunc 17324 funcpropd 17361 natfval 17407 natpropd 17439 dprdval 19344 ptval 22421 dfac14 22469 ptuncnv 22658 ptunhmeo 22659 hoidmvle 43756 hoimbl 43787 |
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