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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdcdeq | GIF version |
Description: Conditional equality of a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.) |
Ref | Expression |
---|---|
bdcdeq.1 | ⊢ BOUNDED 𝜑 |
Ref | Expression |
---|---|
bdcdeq | ⊢ BOUNDED CondEq(𝑥 = 𝑦 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-bdeq 9940 | . . 3 ⊢ BOUNDED 𝑥 = 𝑦 | |
2 | bdcdeq.1 | . . 3 ⊢ BOUNDED 𝜑 | |
3 | 1, 2 | ax-bdim 9934 | . 2 ⊢ BOUNDED (𝑥 = 𝑦 → 𝜑) |
4 | df-cdeq 2748 | . 2 ⊢ (CondEq(𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑)) | |
5 | 3, 4 | bd0r 9945 | 1 ⊢ BOUNDED CondEq(𝑥 = 𝑦 → 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 CondEqwcdeq 2747 BOUNDED wbd 9932 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-bd0 9933 ax-bdim 9934 ax-bdeq 9940 |
This theorem depends on definitions: df-bi 110 df-cdeq 2748 |
This theorem is referenced by: (None) |
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