Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-lem-moexsb | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
wl-lem-moexsb | ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → (∃𝑥𝜑 ↔ [𝑧 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfa1 2156 | . . 3 ⊢ Ⅎ𝑥∀𝑥(𝜑 → 𝑥 = 𝑧) | |
2 | nfs1v 2161 | . . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑 | |
3 | sp 2184 | . . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → (𝜑 → 𝑥 = 𝑧)) | |
4 | ax12v2 2181 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) | |
5 | 3, 4 | syli 39 | . . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) |
6 | sb6 2095 | . . . 4 ⊢ ([𝑧 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑧 → 𝜑)) | |
7 | 5, 6 | syl6ibr 255 | . . 3 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → (𝜑 → [𝑧 / 𝑥]𝜑)) |
8 | 1, 2, 7 | exlimd 2220 | . 2 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → (∃𝑥𝜑 → [𝑧 / 𝑥]𝜑)) |
9 | spsbe 2092 | . 2 ⊢ ([𝑧 / 𝑥]𝜑 → ∃𝑥𝜑) | |
10 | 8, 9 | impbid1 228 | 1 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → (∃𝑥𝜑 ↔ [𝑧 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1540 ∃wex 1786 [wsb 2074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-10 2145 ax-12 2179 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-ex 1787 df-nf 1791 df-sb 2075 |
This theorem is referenced by: (None) |
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