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Mirrors > Home > ILE Home > Th. List > ax11v | GIF version |
Description: This is a version of ax-11o 1701 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 1583 | . 2 ⊢ ∃z z = y | |
2 | ax-17 1416 | . . . . 5 ⊢ (φ → ∀zφ) | |
3 | ax-11 1394 | . . . . 5 ⊢ (x = z → (∀zφ → ∀x(x = z → φ))) | |
4 | 2, 3 | syl5 28 | . . . 4 ⊢ (x = z → (φ → ∀x(x = z → φ))) |
5 | equequ2 1596 | . . . . 5 ⊢ (z = y → (x = z ↔ x = y)) | |
6 | 5 | imbi1d 220 | . . . . . . 7 ⊢ (z = y → ((x = z → φ) ↔ (x = y → φ))) |
7 | 6 | albidv 1702 | . . . . . 6 ⊢ (z = y → (∀x(x = z → φ) ↔ ∀x(x = y → φ))) |
8 | 7 | imbi2d 219 | . . . . 5 ⊢ (z = y → ((φ → ∀x(x = z → φ)) ↔ (φ → ∀x(x = y → φ)))) |
9 | 5, 8 | imbi12d 223 | . . . 4 ⊢ (z = y → ((x = z → (φ → ∀x(x = z → φ))) ↔ (x = y → (φ → ∀x(x = y → φ))))) |
10 | 4, 9 | mpbii 136 | . . 3 ⊢ (z = y → (x = y → (φ → ∀x(x = y → φ)))) |
11 | 10 | exlimiv 1486 | . 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 1240 = wceq 1242 ∃wex 1378 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-gen 1335 ax-ie2 1380 ax-8 1392 ax-11 1394 ax-17 1416 ax-i9 1420 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: equs5or 1708 sb56 1762 |
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