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Mirrors > Home > MPE Home > Th. List > cdeqim | Structured version Visualization version GIF version |
Description: Distribute conditional equality over implication. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Ref | Expression |
---|---|
cdeqnot.1 | ⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
cdeqim.1 | ⊢ CondEq(𝑥 = 𝑦 → (𝜒 ↔ 𝜃)) |
Ref | Expression |
---|---|
cdeqim | ⊢ CondEq(𝑥 = 𝑦 → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdeqnot.1 | . . . 4 ⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
2 | 1 | cdeqri 3388 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
3 | cdeqim.1 | . . . 4 ⊢ CondEq(𝑥 = 𝑦 → (𝜒 ↔ 𝜃)) | |
4 | 3 | cdeqri 3388 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜒 ↔ 𝜃)) |
5 | 2, 4 | imbi12d 333 | . 2 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃))) |
6 | 5 | cdeqi 3387 | 1 ⊢ CondEq(𝑥 = 𝑦 → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 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: (None) |
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