Theorem abvALT 3442

Description: Alternate proof of abv 3441, shorter but using more axioms. (Contributed by BJ, 19-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
abvALT	⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑥𝜑)

Proof of Theorem abvALT

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-clab 2717	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
2	1	albii 1825	. 2 ⊢ (∀𝑦 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
3		eqv 3439	. 2 ⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑦 𝑦 ∈ {𝑥 ∣ 𝜑})
4		nfv 1920	. . 3 ⊢ Ⅎ𝑦𝜑
5	4	sb8v 2353	. 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
6	2, 3, 5	3bitr4i 302	1 ⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑥𝜑)

Syntax hints:   ↔ wb 205  ∀wal 1539   = wceq 1541  [wsb 2070   ∈ wcel 2109  {cab 2716  Vcvv 3430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1544  df-ex 1786  df-nf 1790  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-v 3432

This theorem is referenced by: (None)