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Theorem wl-lem-moexsb 34798
Description: The antecedent 𝑥(𝜑𝑥 = 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 2151 . . 3 𝑥𝑥(𝜑𝑥 = 𝑧)
2 nfs1v 2269 . . 3 𝑥[𝑧 / 𝑥]𝜑
3 sp 2177 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑧) → (𝜑𝑥 = 𝑧))
4 ax12v2 2174 . . . . 5 (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))
53, 4syli 39 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))
6 sb6 2089 . . . 4 ([𝑧 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑧𝜑))
75, 6syl6ibr 254 . . 3 (∀𝑥(𝜑𝑥 = 𝑧) → (𝜑 → [𝑧 / 𝑥]𝜑))
81, 2, 7exlimd 2213 . 2 (∀𝑥(𝜑𝑥 = 𝑧) → (∃𝑥𝜑 → [𝑧 / 𝑥]𝜑))
9 spsbe 2084 . 2 ([𝑧 / 𝑥]𝜑 → ∃𝑥𝜑)
108, 9impbid1 227 1 (∀𝑥(𝜑𝑥 = 𝑧) → (∃𝑥𝜑 ↔ [𝑧 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wal 1531  wex 1776  [wsb 2065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-10 2141  ax-12 2172
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1777  df-nf 1781  df-sb 2066
This theorem is referenced by: (None)
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