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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 34778 (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 3400 | . 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥} | |
2 | equid 2022 | . . . 4 ⊢ 𝑥 = 𝑥 | |
3 | sa-abvi.1 | . . . 4 ⊢ 𝜑 | |
4 | 2, 3 | 2th 267 | . . 3 ⊢ (𝑥 = 𝑥 ↔ 𝜑) |
5 | 4 | abbii 2801 | . 2 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑥 ∣ 𝜑} |
6 | 1, 5 | eqtri 2759 | 1 ⊢ V = {𝑥 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 {cab 2714 Vcvv 3398 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-v 3400 |
This theorem is referenced by: (None) |
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