Theorem abvALT 3494

Description: Alternate proof of abv 3493, 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 2715	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
2	1	albii 1818	. 2 ⊢ (∀𝑦 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
3		eqv 3491	. 2 ⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑦 𝑦 ∈ {𝑥 ∣ 𝜑})
4		sb8v 2355	. 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
5	2, 3, 4	3bitr4i 303	1 ⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑥𝜑)

Syntax hints:   ↔ wb 206  ∀wal 1537   = wceq 1539  [wsb 2064   ∈ wcel 2108  {cab 2714  Vcvv 3481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3483

This theorem is referenced by: (None)