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Theorem wl-dfclab 35273
 Description: Rederive df-clab 2736 from wl-clabv 35272. (Contributed by Wolf Lammen, 31-May-2023.) (Proof modification is discouraged.)
Assertion
Ref Expression
wl-dfclab (𝑥 ∈ {𝑦𝜑} ↔ [𝑥 / 𝑦]𝜑)

Proof of Theorem wl-dfclab
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dfclel 2831 . 2 (𝑥 ∈ {𝑦𝜑} ↔ ∃𝑧(𝑧 = 𝑥𝑧 ∈ {𝑦𝜑}))
2 wl-clabv 35272 . . . . 5 (𝑧 ∈ {𝑦𝜑} ↔ [𝑧 / 𝑦]𝜑)
3 sbequ 2088 . . . . 5 (𝑧 = 𝑥 → ([𝑧 / 𝑦]𝜑 ↔ [𝑥 / 𝑦]𝜑))
42, 3syl5bb 286 . . . 4 (𝑧 = 𝑥 → (𝑧 ∈ {𝑦𝜑} ↔ [𝑥 / 𝑦]𝜑))
54pm5.32i 578 . . 3 ((𝑧 = 𝑥𝑧 ∈ {𝑦𝜑}) ↔ (𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑))
65exbii 1849 . 2 (∃𝑧(𝑧 = 𝑥𝑧 ∈ {𝑦𝜑}) ↔ ∃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
98biantrur 534 . . 3 ([𝑥 / 𝑦]𝜑 ↔ (∃𝑧 𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑))
107, 9bitr4i 281 . 2 (∃𝑧(𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑) ↔ [𝑥 / 𝑦]𝜑)
111, 6, 103bitri 300 1 (𝑥 ∈ {𝑦𝜑} ↔ [𝑥 / 𝑦]𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399  ∃wex 1781  [wsb 2069   ∈ wcel 2111  {cab 2735 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2736  df-clel 2830 This theorem is referenced by: (None)
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