Theorem wl-dfclab 38100

Description: Rederive df-clab 2744 from wl-clabv 38099. (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 2841	. 2 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ {𝑦 ∣ 𝜑}))
2		wl-clabv 38099	. . . . 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 584	. . 3 ⊢ ((𝑧 = 𝑥 ∧ 𝑧 ∈ {𝑦 ∣ 𝜑}) ↔ (𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑))
6	5	exbii 1871	. 2 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ {𝑦 ∣ 𝜑}) ↔ ∃𝑧(𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑))
7		19.41v 1972	. . 3 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑) ↔ (∃𝑧 𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑))
8		ax6ev 1992	. . . 4 ⊢ ∃𝑧 𝑧 = 𝑥
9	8	biantrur 539	. . 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)

Syntax hints:   ↔ wb 209   ∧ wa 400  ∃wex 1802  [wsb 2093   ∈ wcel 2145  {cab 2743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1803  df-sb 2094  df-clab 2744  df-clel 2840

This theorem is referenced by: (None)