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Mirrors > Home > MPE Home > Th. List > Mathboxes > ax5eq | Structured version Visualization version GIF version |
Description: Theorem to add distinct quantifier to atomic formula. (This theorem demonstrates the induction basis for ax-5 1869 considered as a metatheorem. Do not use it for later proofs - use ax-5 1869 instead, to avoid reference to the redundant axiom ax-c16 35473.) (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ax5eq | ⊢ (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-c9 35471 | . 2 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) | |
2 | ax-c16 35473 | . 2 ⊢ (∀𝑧 𝑧 = 𝑥 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) | |
3 | ax-c16 35473 | . 2 ⊢ (∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) | |
4 | 1, 2, 3 | pm2.61ii 178 | 1 ⊢ (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1505 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-c9 35471 ax-c16 35473 |
This theorem is referenced by: dveeq1-o16 35517 |
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