Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2267) or a non-freeness hypothesis (sb5f 2531). See also sb1v 2086. (Contributed by NM, 13-May-1993.) Revise df-sb 2061. (Revised by Wolf Lammen, 21-Feb-2024.) |
Ref | Expression |
---|---|
sb1 | ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spsbe 2079 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥𝜑) | |
2 | pm3.2 470 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → (𝑥 = 𝑦 ∧ 𝜑))) | |
3 | 2 | aleximi 1823 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
4 | 1, 3 | syl5 34 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
5 | sb3b 2494 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) | |
6 | 5 | biimpd 230 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
7 | 4, 6 | pm2.61i 183 | 1 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∀wal 1526 ∃wex 1771 [wsb 2060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-12 2167 ax-13 2381 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ex 1772 df-nf 1776 df-sb 2061 |
This theorem is referenced by: sb3bOLD 2501 dfsb1 2503 spsbeOLDOLD 2504 sb4vOLDOLD 2506 sb4e 2517 |
Copyright terms: Public domain | W3C validator |