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Mirrors > Home > MPE Home > Th. List > cdeqi | Structured version Visualization version 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 3773 | . 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 3772 |
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 3773 |
This theorem is referenced by: cdeqth 3776 cdeqnot 3777 cdeqal 3778 cdeqab 3779 cdeqim 3782 cdeqcv 3783 cdeqeq 3784 cdeqel 3785 |
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