Theorem abv 3469

Description: The class of sets verifying a property is the universal class if and only if that property is a tautology. The reverse implication (bj-abv 37600) requires fewer axioms. (Contributed by BJ, 19-Mar-2021.) Avoid df-clel 2840, ax-8 2148. (Revised by GG, 30-Aug-2024.) (Proof shortened by BJ, 30-Aug-2024.)

Assertion

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

Proof of Theorem abv

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfcleq 2758	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ ⊤} ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊤}))
2		vextru 2750	. . . . . 6 ⊢ 𝑦 ∈ {𝑥 ∣ ⊤}
3	2	tbt 372	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊤}))
4		df-clab 2744	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
5	3, 4	bitr3i 280	. . . 4 ⊢ ((𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊤}) ↔ [𝑦 / 𝑥]𝜑)
6	5	albii 1852	. . 3 ⊢ (∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊤}) ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
7	1, 6	bitri 278	. 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ ⊤} ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
8		dfv2 3460	. . 3 ⊢ V = {𝑥 ∣ ⊤}
9	8	eqeq2i 2778	. 2 ⊢ ({𝑥 ∣ 𝜑} = V ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊤})
10		sb8v 2387	. 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
11	7, 9, 10	3bitr4i 306	1 ⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑥𝜑)

This proof depends on syntax axioms:   ↔ wb 209  ∀wal 1568   = wceq 1570  ⊤wtru 1571  [wsb 2099   ∈ wcel 2146  {cab 2743  Vcvv 3457

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-9 2156  ax-11 2195  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-v 3459

This theorem is used by:  dfnf5  4338  mh-setind  37106  ecqmap  39158