Theorem vexw 2805

Description: If 𝜑 is a theorem, then any set belongs to the class {𝑥 ∣ 𝜑}. Therefore, {𝑥 ∣ 𝜑} is "a" universal class.

This is the closest one can get to defining a universal class, or proving vex 3497, without using ax-ext 2793. Note that this theorem has no disjoint variable condition and does not use df-clel 2893 nor df-cleq 2814 either: only propositional logic and ax-gen 1796 and df-clab 2800. This is sbt 2071 expressed using class abstractions.

Without ax-ext 2793, one cannot define "the" universal class, since one could not prove for instance the justification theorem {𝑥 ∣ ⊤} = {𝑦 ∣ ⊤} (see vjust 3495). Indeed, in order to prove any equality of classes, one needs df-cleq 2814, which has ax-ext 2793 as a hypothesis. Therefore, the classes {𝑥 ∣ ⊤}, {𝑦 ∣ (𝜑 → 𝜑)}, {𝑧 ∣ (∀𝑡𝑡 = 𝑡 → ∀𝑡𝑡 = 𝑡)} and countless others are all universal classes whose equality cannot be proved without ax-ext 2793. Once dfcleq 2815 is available, we will define "the" universal class in df-v 3496.

Its degenerate instance is also a simple consequence of abid 2803 (using mpbir 233). (Contributed by BJ, 13-Jun-2019.) Reduce axiom dependencies. (Revised by Steven Nguyen, 25-Apr-2023.)

Hypothesis

Ref	Expression
vexw.1	⊢ 𝜑

Assertion

Ref	Expression
vexw	⊢ 𝑦 ∈ {𝑥 ∣ 𝜑}

Proof of Theorem vexw

Step	Hyp	Ref	Expression
1		vexw.1	. . 3 ⊢ 𝜑
2	1	sbt 2071	. 2 ⊢ [𝑦 / 𝑥]𝜑
3		df-clab 2800	. 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
4	2, 3	mpbir 233	1 ⊢ 𝑦 ∈ {𝑥 ∣ 𝜑}

Syntax hints:  [wsb 2069   ∈ wcel 2114  {cab 2799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796

This theorem depends on definitions:  df-bi 209  df-sb 2070  df-clab 2800

This theorem is referenced by:  vextru  2806  vex  3497  bj-ralvw  34198  bj-rexvw  34199  bj-rababw  34200  bj-df-v  34350