Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 3386 | . 2 ⊢ (CondEq(𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑)) | |
3 | 1, 2 | mpbir 220 | 1 ⊢ CondEq(𝑥 = 𝑦 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 CondEqwcdeq 3385 |
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 196 df-cdeq 3386 |
This theorem is referenced by: cdeqth 3389 cdeqnot 3390 cdeqal 3391 cdeqab 3392 cdeqim 3395 cdeqcv 3396 cdeqeq 3397 cdeqel 3398 bj-cdeqab 31975 |
Copyright terms: Public domain | W3C validator |