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| 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 3769 | . 2 ⊢ (CondEq(𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑)) | |
| 3 | 1, 2 | mpbir 231 | 1 ⊢ CondEq(𝑥 = 𝑦 → 𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 CondEqwcdeq 3768 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 | 
| This theorem depends on definitions: df-bi 207 df-cdeq 3769 | 
| This theorem is referenced by: cdeqth 3772 cdeqnot 3773 cdeqal 3774 cdeqab 3775 cdeqim 3778 cdeqcv 3779 cdeqeq 3780 cdeqel 3781 | 
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