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Theorem wl-dfclab 36077
Description: Rederive df-clab 2715 from wl-clabv 36076. (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 2816 . 2 (𝑥 ∈ {𝑦𝜑} ↔ ∃𝑧(𝑧 = 𝑥𝑧 ∈ {𝑦𝜑}))
2 wl-clabv 36076 . . . . 5 (𝑧 ∈ {𝑦𝜑} ↔ [𝑧 / 𝑦]𝜑)
3 sbequ 2087 . . . . 5 (𝑧 = 𝑥 → ([𝑧 / 𝑦]𝜑 ↔ [𝑥 / 𝑦]𝜑))
42, 3bitrid 283 . . . 4 (𝑧 = 𝑥 → (𝑧 ∈ {𝑦𝜑} ↔ [𝑥 / 𝑦]𝜑))
54pm5.32i 576 . . 3 ((𝑧 = 𝑥𝑧 ∈ {𝑦𝜑}) ↔ (𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑))
65exbii 1851 . 2 (∃𝑧(𝑧 = 𝑥𝑧 ∈ {𝑦𝜑}) ↔ ∃𝑧(𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑))
7 19.41v 1954 . . 3 (∃𝑧(𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑) ↔ (∃𝑧 𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑))
8 ax6ev 1974 . . . 4 𝑧 𝑧 = 𝑥
98biantrur 532 . . 3 ([𝑥 / 𝑦]𝜑 ↔ (∃𝑧 𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑))
107, 9bitr4i 278 . 2 (∃𝑧(𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑) ↔ [𝑥 / 𝑦]𝜑)
111, 6, 103bitri 297 1 (𝑥 ∈ {𝑦𝜑} ↔ [𝑥 / 𝑦]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  wex 1782  [wsb 2068  wcel 2107  {cab 2714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-sb 2069  df-clab 2715  df-clel 2815
This theorem is referenced by: (None)
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