Theorem wl-clabt 37578

Description: Using class abstraction in a context. For a version based on fewer axioms see wl-clabtv 37577. (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 360	. . . 4 ⊢ (𝜑 → (𝜓 ↔ (𝜑 → 𝜓)))
3	1, 2	sbbid 2243	. . 3 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥](𝜑 → 𝜓)))
4		df-clab 2712	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ [𝑦 / 𝑥]𝜓)
5		df-clab 2712	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ (𝜑 → 𝜓)} ↔ [𝑦 / 𝑥](𝜑 → 𝜓))
6	3, 4, 5	3bitr4g 314	. 2 ⊢ (𝜑 → (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ 𝑦 ∈ {𝑥 ∣ (𝜑 → 𝜓)}))
7	6	eqrdv 2732	1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ (𝜑 → 𝜓)})

Syntax hints:   → wi 4   = wceq 1536  Ⅎwnf 1779  [wsb 2061   ∈ wcel 2105  {cab 2711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-9 2115  ax-12 2174  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726

This theorem is referenced by: (None)