Theorem wl-dfclab 38610

Description: Rederive df-clab 2739 from wl-clabv 38609. (Contributed by Wolf Lammen, 31-May-2023.) (Proof modification is discouraged.)

Assertion

Ref	Expression
wl-dfclab	⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)

Proof of Theorem wl-dfclab

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfclel 2836	. 2 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ {𝑦 ∣ 𝜑}))
2		wl-clabv 38609	. . . . 5 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑)
3		sbequ 2119	. . . . 5 ⊢ (𝑧 = 𝑥 → ([𝑧 / 𝑦]𝜑 ↔ [𝑥 / 𝑦]𝜑))
4	2, 3	bitrid 286	. . . 4 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑))
5	4	pm5.32i 585	. . 3 ⊢ ((𝑧 = 𝑥 ∧ 𝑧 ∈ {𝑦 ∣ 𝜑}) ↔ (𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑))
6	5	exbii 1880	. 2 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ {𝑦 ∣ 𝜑}) ↔ ∃𝑧(𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑))
7		19.41v 1981	. . 3 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑) ↔ (∃𝑧 𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑))
8		ax6ev 2001	. . . 4 ⊢ ∃𝑧 𝑧 = 𝑥
9	8	biantrur 540	. . 3 ⊢ ([𝑥 / 𝑦]𝜑 ↔ (∃𝑧 𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑))
10	7, 9	bitr4i 281	. 2 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑) ↔ [𝑥 / 𝑦]𝜑)
11	1, 6, 10	3bitri 300	1 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  ∃wex 1811  [wsb 2098   ∈ wcel 2144  {cab 2738

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2099  df-clab 2739  df-clel 2835

This theorem is used by: (None)