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Mirrors > Home > MPE Home > Th. List > sb1 | Structured version Visualization version GIF version |
Description: One direction of a simplified definition of substitution. The converse requires either a disjoint variable condition (sb5 2205) or a non-freeness hypothesis (sb5f 2459). (Contributed by NM, 13-May-1993.) Revise df-sb 2016. (Revised by Wolf Lammen, 29-Jul-2023.) |
Ref | Expression |
---|---|
sb1 | ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbequ2 2177 | . . . 4 ⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → 𝜑)) | |
2 | 19.8a 2109 | . . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | |
3 | 2 | ex 405 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
4 | 1, 3 | syld 47 | . . 3 ⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
5 | 4 | sps 2113 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
6 | sb4b 2423 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
7 | equs4 2351 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | |
8 | 6, 7 | syl6bi 245 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
9 | 5, 8 | pm2.61i 177 | 1 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 ∀wal 1505 ∃wex 1742 [wsb 2015 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-10 2079 ax-12 2106 ax-13 2301 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-ex 1743 df-nf 1747 df-sb 2016 |
This theorem is referenced by: sb3b 2428 dfsb1 2430 spsbeOLDOLD 2431 sb4vOLDOLD 2433 sb4e 2445 |
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