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| Mirrors > Home > MPE Home > Th. List > Mathboxes > sa-abvi | Structured version Visualization version GIF version | ||
| Description: A theorem about the universal class. Inference associated with bj-abv 36907 (which is proved from fewer axioms). (Contributed by Stefan Allan, 9-Dec-2008.) |
| Ref | Expression |
|---|---|
| sa-abvi.1 | ⊢ 𝜑 |
| Ref | Expression |
|---|---|
| sa-abvi | ⊢ V = {𝑥 ∣ 𝜑} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-v 3482 | . 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥} | |
| 2 | equid 2011 | . . . 4 ⊢ 𝑥 = 𝑥 | |
| 3 | sa-abvi.1 | . . . 4 ⊢ 𝜑 | |
| 4 | 2, 3 | 2th 264 | . . 3 ⊢ (𝑥 = 𝑥 ↔ 𝜑) |
| 5 | 4 | abbii 2809 | . 2 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑥 ∣ 𝜑} |
| 6 | 1, 5 | eqtri 2765 | 1 ⊢ V = {𝑥 ∣ 𝜑} |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 {cab 2714 Vcvv 3480 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-v 3482 |
| This theorem is referenced by: (None) |
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