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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 7919 | . 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 1483 Xcixp 7908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-fn 5891 df-ixp 7909 |
This theorem is referenced by: elixpsn 7947 ixpsnf1o 7948 dfac9 8958 prdsval 16115 isfunc 16524 funcpropd 16560 natfval 16606 natpropd 16636 dprdval 18402 ptval 21373 dfac14 21421 ptuncnv 21610 ptunhmeo 21611 hoidmvle 40814 hoimbl 40845 |
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