Theorem bj-abvALT 37574

Description: Alternate version of bj-abv 37573; shorter but uses ax-8 2148. (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 1943	. . 3 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
2		vexwt 2749	. . 3 ⊢ (∀𝑥𝜑 → 𝑦 ∈ {𝑥 ∣ 𝜑})
3	1, 2	alrimih 1857	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦 𝑦 ∈ {𝑥 ∣ 𝜑})
4		eqv 3468	. 2 ⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑦 𝑦 ∈ {𝑥 ∣ 𝜑})
5	3, 4	sylibr 237	1 ⊢ (∀𝑥𝜑 → {𝑥 ∣ 𝜑} = V)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568   = wceq 1570   ∈ wcel 2146  {cab 2744  Vcvv 3458

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-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-v 3460

This theorem is used by: (None)