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Theorem wl-lem-moexsb 33947
 Description: The antecedent ∀𝑥(𝜑 → 𝑥 = 𝑧) relates to ∃*𝑥𝜑, but is better suited for usage in proofs. Note that no distinct variable restriction is placed on 𝜑. This theorem provides a basic working step in proving theorems about ∃* or ∃!. (Contributed by Wolf Lammen, 3-Oct-2019.)
Assertion
Ref Expression
wl-lem-moexsb (∀𝑥(𝜑𝑥 = 𝑧) → (∃𝑥𝜑 ↔ [𝑧 / 𝑥]𝜑))
Distinct variable group:   𝑥,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑧)

Proof of Theorem wl-lem-moexsb
StepHypRef Expression
1 nfa1 2145 . . 3 𝑥𝑥(𝜑𝑥 = 𝑧)
2 nfs1v 2254 . . 3 𝑥[𝑧 / 𝑥]𝜑
3 sp 2167 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑧) → (𝜑𝑥 = 𝑧))
4 ax12v2 2165 . . . . 5 (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))
53, 4syli 39 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))
6 sb2v 2243 . . . 4 (∀𝑥(𝑥 = 𝑧𝜑) → [𝑧 / 𝑥]𝜑)
75, 6syl6 35 . . 3 (∀𝑥(𝜑𝑥 = 𝑧) → (𝜑 → [𝑧 / 𝑥]𝜑))
81, 2, 7exlimd 2204 . 2 (∀𝑥(𝜑𝑥 = 𝑧) → (∃𝑥𝜑 → [𝑧 / 𝑥]𝜑))
9 spsbe 2015 . 2 ([𝑧 / 𝑥]𝜑 → ∃𝑥𝜑)
108, 9impbid1 217 1 (∀𝑥(𝜑𝑥 = 𝑧) → (∃𝑥𝜑 ↔ [𝑧 / 𝑥]𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1599  ∃wex 1823  [wsb 2011 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-10 2135  ax-12 2163 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-ex 1824  df-nf 1828  df-sb 2012 This theorem is referenced by: (None)
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