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Mirrors > Home > ILE Home > Th. List > cdeqeq | GIF version |
Description: Distribute conditional equality over equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Ref | Expression |
---|---|
cdeqeq.1 | ⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵) |
cdeqeq.2 | ⊢ CondEq(𝑥 = 𝑦 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
cdeqeq | ⊢ CondEq(𝑥 = 𝑦 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdeqeq.1 | . . . 4 ⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵) | |
2 | 1 | cdeqri 2932 | . . 3 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) |
3 | cdeqeq.2 | . . . 4 ⊢ CondEq(𝑥 = 𝑦 → 𝐶 = 𝐷) | |
4 | 3 | cdeqri 2932 | . . 3 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷) |
5 | 2, 4 | eqeq12d 2179 | . 2 ⊢ (𝑥 = 𝑦 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) |
6 | 5 | cdeqi 2931 | 1 ⊢ CondEq(𝑥 = 𝑦 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1342 CondEqwcdeq 2929 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1434 ax-gen 1436 ax-4 1497 ax-17 1513 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-cleq 2157 df-cdeq 2930 |
This theorem is referenced by: (None) |
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