Theorem foo3 28686

Description: A theorem about the universal class. (Contributed by Stefan Allan, 9-Dec-2008.)

Hypothesis

Ref	Expression
foo3.1	⊢ 𝜑

Assertion

Ref	Expression
foo3	⊢ V = {𝑥 ∣ 𝜑}

Proof of Theorem foo3

Step	Hyp	Ref	Expression
1		df-v 3175	. 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
2		equid 1926	. . . 4 ⊢ 𝑥 = 𝑥
3		foo3.1	. . . 4 ⊢ 𝜑
4	2, 3	2th 253	. . 3 ⊢ (𝑥 = 𝑥 ↔ 𝜑)
5	4	abbii 2726	. 2 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑥 ∣ 𝜑}
6	1, 5	eqtri 2632	1 ⊢ V = {𝑥 ∣ 𝜑}

Syntax hints:   = wceq 1475  {cab 2596  Vcvv 3173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175

This theorem is referenced by: (None)