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Mirrors > Home > MPE Home > Th. List > cdeqel | Structured version Visualization version GIF version |
Description: Distribute conditional equality over elementhood. (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 |
---|---|
cdeqel | ⊢ CondEq(𝑥 = 𝑦 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdeqeq.1 | . . . 4 ⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵) | |
2 | 1 | cdeqri 3756 | . . 3 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) |
3 | cdeqeq.2 | . . . 4 ⊢ CondEq(𝑥 = 𝑦 → 𝐶 = 𝐷) | |
4 | 3 | cdeqri 3756 | . . 3 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷) |
5 | 2, 4 | eleq12d 2907 | . 2 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)) |
6 | 5 | cdeqi 3755 | 1 ⊢ CondEq(𝑥 = 𝑦 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1533 ∈ wcel 2110 CondEqwcdeq 3753 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1777 df-cleq 2814 df-clel 2893 df-cdeq 3754 |
This theorem is referenced by: nfccdeq 3768 |
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