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Mirrors > Home > MPE Home > Th. List > cdeqri | Structured version Visualization version 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 3419 | . 2 ⊢ (CondEq(𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑)) | |
3 | 1, 2 | mpbi 220 | 1 ⊢ (𝑥 = 𝑦 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 CondEqwcdeq 3418 |
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 197 df-cdeq 3419 |
This theorem is referenced by: cdeqnot 3423 cdeqal 3424 cdeqab 3425 cdeqal1 3426 cdeqab1 3427 cdeqim 3428 cdeqeq 3430 cdeqel 3431 nfcdeq 3432 bj-cdeqab 32787 |
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