Theorem wl-clabt 37971

Description: Using class abstraction in a context. For a version based on fewer axioms see wl-clabtv 37970. (Contributed by Wolf Lammen, 29-May-2023.)

Hypothesis

Ref	Expression
wl-clabt.nf	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
wl-clabt	⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ (𝜑 → 𝜓)})

Proof of Theorem wl-clabt

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wl-clabt.nf	. . . 4 ⊢ Ⅎ𝑥𝜑
2		biimt 362	. . . 4 ⊢ (𝜑 → (𝜓 ↔ (𝜑 → 𝜓)))
3	1, 2	sbbid 2260	. . 3 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥](𝜑 → 𝜓)))
4		df-clab 2720	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ [𝑦 / 𝑥]𝜓)
5		df-clab 2720	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ (𝜑 → 𝜓)} ↔ [𝑦 / 𝑥](𝜑 → 𝜓))
6	3, 4, 5	3bitr4g 316	. 2 ⊢ (𝜑 → (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ 𝑦 ∈ {𝑥 ∣ (𝜑 → 𝜓)}))
7	6	eqrdv 2739	1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ (𝜑 → 𝜓)})

Syntax hints:   → wi 4   = wceq 1548  Ⅎwnf 1791  [wsb 2074   ∈ wcel 2121  {cab 2719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-9 2131  ax-12 2191  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-ex 1788  df-nf 1792  df-sb 2075  df-clab 2720  df-cleq 2733

This theorem is referenced by: (None)