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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 8947 | . 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 1537 Xcixp 8936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-fn 6566 df-ixp 8937 |
This theorem is referenced by: elixpsn 8976 ixpsnf1o 8977 dfac9 10175 prdsval 17502 isfunc 17915 funcpropd 17954 natfval 18001 natpropd 18033 dprdval 20038 ptval 23594 dfac14 23642 ptuncnv 23831 ptunhmeo 23832 hoidmvle 46556 hoimbl 46587 |
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