Theorem bj-abvALT 37508

Description: Alternate version of bj-abv 37507; shorter but uses ax-8 2143. (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-abvALT	⊢ (∀𝑥𝜑 → {𝑥 ∣ 𝜑} = V)

Proof of Theorem bj-abvALT

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-5 1938	. . 3 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
2		vexwt 2744	. . 3 ⊢ (∀𝑥𝜑 → 𝑦 ∈ {𝑥 ∣ 𝜑})
3	1, 2	alrimih 1852	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦 𝑦 ∈ {𝑥 ∣ 𝜑})
4		eqv 3463	. 2 ⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑦 𝑦 ∈ {𝑥 ∣ 𝜑})
5	3, 4	sylibr 237	1 ⊢ (∀𝑥𝜑 → {𝑥 ∣ 𝜑} = V)

Syntax hints:   → wi 4  ∀wal 1566   = wceq 1568   ∈ wcel 2141  {cab 2739  Vcvv 3453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455

This theorem is referenced by: (None)