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Mirrors > Home > ILE Home > Th. List > equsex | GIF version |
Description: A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.) |
Ref | Expression |
---|---|
equsex.1 | ⊢ (𝜓 → ∀𝑥𝜓) |
equsex.2 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
equsex | ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equsex.1 | . . 3 ⊢ (𝜓 → ∀𝑥𝜓) | |
2 | equsex.2 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 2 | biimpa 290 | . . 3 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → 𝜓) |
4 | 1, 3 | exlimih 1524 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → 𝜓) |
5 | a9e 1626 | . . 3 ⊢ ∃𝑥 𝑥 = 𝑦 | |
6 | idd 21 | . . . . 5 ⊢ (𝜓 → (𝑥 = 𝑦 → 𝑥 = 𝑦)) | |
7 | 2 | biimprcd 158 | . . . . 5 ⊢ (𝜓 → (𝑥 = 𝑦 → 𝜑)) |
8 | 6, 7 | jcad 301 | . . . 4 ⊢ (𝜓 → (𝑥 = 𝑦 → (𝑥 = 𝑦 ∧ 𝜑))) |
9 | 1, 8 | eximdh 1542 | . . 3 ⊢ (𝜓 → (∃𝑥 𝑥 = 𝑦 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
10 | 5, 9 | mpi 15 | . 2 ⊢ (𝜓 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
11 | 4, 10 | impbii 124 | 1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∀wal 1282 = wceq 1284 ∃wex 1421 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-4 1440 ax-i9 1463 ax-ial 1467 |
This theorem depends on definitions: df-bi 115 |
This theorem is referenced by: cbvexh 1678 sb56 1806 cleljust 1854 sb10f 1912 |
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