New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > ax17eq | GIF version |
Description: Theorem to add distinct quantifier to atomic formula. (This theorem demonstrates the induction basis for ax-17 1616 considered as a metatheorem. Do not use it for later proofs - use ax-17 1616 instead, to avoid reference to the redundant axiom ax-16 2144.) (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ax17eq | ⊢ (x = y → ∀z x = y) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-12o 2142 | . 2 ⊢ (¬ ∀z z = x → (¬ ∀z z = y → (x = y → ∀z x = y))) | |
2 | ax-16 2144 | . 2 ⊢ (∀z z = x → (x = y → ∀z x = y)) | |
3 | ax-16 2144 | . 2 ⊢ (∀z z = y → (x = y → ∀z x = y)) | |
4 | 1, 2, 3 | pm2.61ii 157 | 1 ⊢ (x = y → ∀z x = y) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-12o 2142 ax-16 2144 |
This theorem is referenced by: dveeq2-o16 2185 dveeq1-o16 2188 |
Copyright terms: Public domain | W3C validator |