Theorem sa-abvi 32866

Description: A theorem about the universal class. Inference associated with bj-abv 37598 (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 3459	. 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
2		equid 2045	. . . 4 ⊢ 𝑥 = 𝑥
3		sa-abvi.1	. . . 4 ⊢ 𝜑
4	2, 3	2th 267	. . 3 ⊢ (𝑥 = 𝑥 ↔ 𝜑)
5	4	abbii 2832	. 2 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑥 ∣ 𝜑}
6	1, 5	eqtri 2788	1 ⊢ V = {𝑥 ∣ 𝜑}

This proof depends on syntax axioms:   = wceq 1570  {cab 2743  Vcvv 3457

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-v 3459

This theorem is used by: (None)