Theorem foo3 29302

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 3202	. 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
2		equid 1939	. . . 4 ⊢ 𝑥 = 𝑥
3		foo3.1	. . . 4 ⊢ 𝜑
4	2, 3	2th 254	. . 3 ⊢ (𝑥 = 𝑥 ↔ 𝜑)
5	4	abbii 2739	. 2 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑥 ∣ 𝜑}
6	1, 5	eqtri 2644	1 ⊢ V = {𝑥 ∣ 𝜑}

Syntax hints:   = wceq 1483  {cab 2608  Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by: (None)