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Mirrors > Home > ILE Home > Th. List > axi12 | GIF version |
Description: Proof that ax-i12 1395 follows from ax-bndl 1396. So that we can track which theorems rely on ax-bndl 1396, proofs should reference ax-i12 1395 rather than this theorem. (Contributed by Jim Kingdon, 17-Aug-2018.) (New usage is discouraged). (Proof modification is discouraged.) |
Ref | Expression |
---|---|
axi12 | ⊢ (∀z z = x ∨ (∀z z = y ∨ ∀z(x = y → ∀z x = y))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-bndl 1396 | . 2 ⊢ (∀z z = x ∨ (∀z z = y ∨ ∀x∀z(x = y → ∀z x = y))) | |
2 | sp 1398 | . . . 4 ⊢ (∀x∀z(x = y → ∀z x = y) → ∀z(x = y → ∀z x = y)) | |
3 | 2 | orim2i 677 | . . 3 ⊢ ((∀z z = y ∨ ∀x∀z(x = y → ∀z x = y)) → (∀z z = y ∨ ∀z(x = y → ∀z x = y))) |
4 | 3 | orim2i 677 | . 2 ⊢ ((∀z z = x ∨ (∀z z = y ∨ ∀x∀z(x = y → ∀z x = y))) → (∀z z = x ∨ (∀z z = y ∨ ∀z(x = y → ∀z x = y)))) |
5 | 1, 4 | ax-mp 7 | 1 ⊢ (∀z z = x ∨ (∀z z = y ∨ ∀z(x = y → ∀z x = y))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 628 ∀wal 1240 = wceq 1242 |
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-io 629 ax-bndl 1396 ax-4 1397 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: (None) |
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