Theorem wl-dfclab 34843

Description: Rederive df-clab 2800 from wl-clabv 34842. (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 2894	. 2 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ {𝑦 ∣ 𝜑}))
2		wl-clabv 34842	. . . . 5 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑)
3		sbequ 2090	. . . . 5 ⊢ (𝑧 = 𝑥 → ([𝑧 / 𝑦]𝜑 ↔ [𝑥 / 𝑦]𝜑))
4	2, 3	syl5bb 285	. . . 4 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑))
5	4	pm5.32i 577	. . 3 ⊢ ((𝑧 = 𝑥 ∧ 𝑧 ∈ {𝑦 ∣ 𝜑}) ↔ (𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑))
6	5	exbii 1848	. 2 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ {𝑦 ∣ 𝜑}) ↔ ∃𝑧(𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑))
7		19.41v 1950	. . 3 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑) ↔ (∃𝑧 𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑))
8		ax6ev 1972	. . . 4 ⊢ ∃𝑧 𝑧 = 𝑥
9	8	biantrur 533	. . 3 ⊢ ([𝑥 / 𝑦]𝜑 ↔ (∃𝑧 𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑))
10	7, 9	bitr4i 280	. 2 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑) ↔ [𝑥 / 𝑦]𝜑)
11	1, 6, 10	3bitri 299	1 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)

Syntax hints:   ↔ wb 208   ∧ wa 398  ∃wex 1780  [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  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070  df-clab 2800  df-clel 2893

This theorem is referenced by: (None)