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Mirrors > Home > ILE Home > Th. List > cdeqi | GIF version |
Description: Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Ref | Expression |
---|---|
cdeqi.1 | ⊢ (𝑥 = 𝑦 → 𝜑) |
Ref | Expression |
---|---|
cdeqi | ⊢ CondEq(𝑥 = 𝑦 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdeqi.1 | . 2 ⊢ (𝑥 = 𝑦 → 𝜑) | |
2 | df-cdeq 2946 | . 2 ⊢ (CondEq(𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑)) | |
3 | 1, 2 | mpbir 146 | 1 ⊢ CondEq(𝑥 = 𝑦 → 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 CondEqwcdeq 2945 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 |
This theorem depends on definitions: df-bi 117 df-cdeq 2946 |
This theorem is referenced by: cdeqth 2949 cdeqnot 2950 cdeqal 2951 cdeqab 2952 cdeqim 2955 cdeqcv 2956 cdeqeq 2957 cdeqel 2958 |
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