Theorem sa-abvi 32131

Description: A theorem about the universal class. Inference associated with bj-abv 36253 (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 3475	. 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
2		equid 2014	. . . 4 ⊢ 𝑥 = 𝑥
3		sa-abvi.1	. . . 4 ⊢ 𝜑
4	2, 3	2th 264	. . 3 ⊢ (𝑥 = 𝑥 ↔ 𝜑)
5	4	abbii 2801	. 2 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑥 ∣ 𝜑}
6	1, 5	eqtri 2759	1 ⊢ V = {𝑥 ∣ 𝜑}

Syntax hints:   = wceq 1540  {cab 2708  Vcvv 3473

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 1912  ax-6 1970  ax-7 2010  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-v 3475

This theorem is referenced by: (None)