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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 33486 (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 3399 | . 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥} | |
2 | equid 2058 | . . . 4 ⊢ 𝑥 = 𝑥 | |
3 | sa-abvi.1 | . . . 4 ⊢ 𝜑 | |
4 | 2, 3 | 2th 256 | . . 3 ⊢ (𝑥 = 𝑥 ↔ 𝜑) |
5 | 4 | abbii 2907 | . 2 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑥 ∣ 𝜑} |
6 | 1, 5 | eqtri 2801 | 1 ⊢ V = {𝑥 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 {cab 2762 Vcvv 3397 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-ext 2753 |
This theorem depends on definitions: df-bi 199 df-an 387 df-tru 1605 df-ex 1824 df-sb 2012 df-clab 2763 df-cleq 2769 df-v 3399 |
This theorem is referenced by: (None) |
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