Theorem sa-abvi 30706

Description: A theorem about the universal class. Inference associated with bj-abv 35018 (which is proved from fewer axioms). (Contributed by Stefan Allan, 9-Dec-2008.)

Hypothesis

Ref	Expression
sa-abvi.1	⊢ 𝜑

Assertion

Ref	Expression
sa-abvi	⊢ V = {𝑥 ∣ 𝜑}

Proof of Theorem sa-abvi

Step	Hyp	Ref	Expression
1		df-v 3424	. 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
2		equid 2016	. . . 4 ⊢ 𝑥 = 𝑥
3		sa-abvi.1	. . . 4 ⊢ 𝜑
4	2, 3	2th 263	. . 3 ⊢ (𝑥 = 𝑥 ↔ 𝜑)
5	4	abbii 2809	. 2 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑥 ∣ 𝜑}
6	1, 5	eqtri 2766	1 ⊢ V = {𝑥 ∣ 𝜑}

Syntax hints:   = wceq 1539  {cab 2715  Vcvv 3422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-v 3424

This theorem is referenced by: (None)