Mathbox for Stefan Allan |
< Previous
Next >
Nearby theorems |
||
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 34225 (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 3498 | . 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥} | |
2 | equid 2019 | . . . 4 ⊢ 𝑥 = 𝑥 | |
3 | sa-abvi.1 | . . . 4 ⊢ 𝜑 | |
4 | 2, 3 | 2th 266 | . . 3 ⊢ (𝑥 = 𝑥 ↔ 𝜑) |
5 | 4 | abbii 2888 | . 2 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑥 ∣ 𝜑} |
6 | 1, 5 | eqtri 2846 | 1 ⊢ V = {𝑥 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 {cab 2801 Vcvv 3496 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-9 2124 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-sb 2070 df-clab 2802 df-cleq 2816 df-v 3498 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |