![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > ax11v | GIF version |
Description: This is a version of ax-11o 1752 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.) (Revised by Jim Kingdon, 15-Dec-2017.) |
Ref | Expression |
---|---|
ax11v | ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | a9e 1632 | . 2 ⊢ ∃𝑧 𝑧 = 𝑦 | |
2 | ax-17 1465 | . . . . 5 ⊢ (𝜑 → ∀𝑧𝜑) | |
3 | ax-11 1443 | . . . . 5 ⊢ (𝑥 = 𝑧 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) | |
4 | 2, 3 | syl5 32 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) |
5 | equequ2 1647 | . . . . 5 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦)) | |
6 | 5 | imbi1d 230 | . . . . . . 7 ⊢ (𝑧 = 𝑦 → ((𝑥 = 𝑧 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑))) |
7 | 6 | albidv 1753 | . . . . . 6 ⊢ (𝑧 = 𝑦 → (∀𝑥(𝑥 = 𝑧 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
8 | 7 | imbi2d 229 | . . . . 5 ⊢ (𝑧 = 𝑦 → ((𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) |
9 | 5, 8 | imbi12d 233 | . . . 4 ⊢ (𝑧 = 𝑦 → ((𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) ↔ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))) |
10 | 4, 9 | mpbii 147 | . . 3 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) |
11 | 10 | exlimiv 1535 | . 2 ⊢ (∃𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) |
12 | 1, 11 | ax-mp 7 | 1 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1288 = wceq 1290 ∃wex 1427 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1382 ax-gen 1384 ax-ie2 1429 ax-8 1441 ax-11 1443 ax-17 1465 ax-i9 1469 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: equs5or 1759 sb56 1814 |
Copyright terms: Public domain | W3C validator |