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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-eqs | Structured version Visualization version GIF version |
Description: A lemma for substitutions, proved from Tarski's FOL. The version without DV (𝑥, 𝑦) is true but requires ax-13 2374. The disjoint variable condition DV (𝑥, 𝜑) is necessary for both directions: consider substituting 𝑥 = 𝑧 for 𝜑. (Contributed by BJ, 25-May-2021.) |
Ref | Expression |
---|---|
bj-eqs | ⊢ (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1 6 | . . 3 ⊢ (𝜑 → (𝑥 = 𝑦 → 𝜑)) | |
2 | 1 | alrimiv 1934 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
3 | exim 1840 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑)) | |
4 | ax6ev 1977 | . . . 4 ⊢ ∃𝑥 𝑥 = 𝑦 | |
5 | pm2.27 42 | . . . 4 ⊢ (∃𝑥 𝑥 = 𝑦 → ((∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑) → ∃𝑥𝜑)) | |
6 | 4, 5 | ax-mp 5 | . . 3 ⊢ ((∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑) → ∃𝑥𝜑) |
7 | ax5e 1919 | . . 3 ⊢ (∃𝑥𝜑 → 𝜑) | |
8 | 3, 6, 7 | 3syl 18 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜑) |
9 | 2, 8 | impbii 208 | 1 ⊢ (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1540 ∃wex 1786 |
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 |
This theorem depends on definitions: df-bi 206 df-ex 1787 |
This theorem is referenced by: bj-sb 34865 |
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