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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 8472 | . 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 8461 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-tru 1540 df-ex 1781 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-ral 3143 df-fn 6358 df-ixp 8462 |
This theorem is referenced by: elixpsn 8501 ixpsnf1o 8502 dfac9 9562 prdsval 16728 isfunc 17134 funcpropd 17170 natfval 17216 natpropd 17246 dprdval 19125 ptval 22178 dfac14 22226 ptuncnv 22415 ptunhmeo 22416 hoidmvle 42902 hoimbl 42933 |
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