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Mirrors > Home > ILE Home > Th. List > cdeqri | GIF version |
Description: Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Ref | Expression |
---|---|
cdeqri.1 | ⊢ CondEq(𝑥 = 𝑦 → 𝜑) |
Ref | Expression |
---|---|
cdeqri | ⊢ (𝑥 = 𝑦 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdeqri.1 | . 2 ⊢ CondEq(𝑥 = 𝑦 → 𝜑) | |
2 | df-cdeq 2897 | . 2 ⊢ (CondEq(𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑)) | |
3 | 1, 2 | mpbi 144 | 1 ⊢ (𝑥 = 𝑦 → 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 CondEqwcdeq 2896 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 |
This theorem depends on definitions: df-bi 116 df-cdeq 2897 |
This theorem is referenced by: cdeqnot 2901 cdeqal 2902 cdeqab 2903 cdeqal1 2904 cdeqab1 2905 cdeqim 2906 cdeqeq 2908 cdeqel 2909 nfcdeq 2910 |
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