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Mirrors > Home > NFE Home > Th. List > ax11i | GIF version |
Description: Inference that has ax-11 1746 (without ∀y) as its conclusion. Uses only Tarski's FOL axiom schemes. The hypotheses may be eliminable without one or more of these axioms in special cases. Proof similar to Lemma 16 of [Tarski] p. 70. (Contributed by NM, 20-May-2008.) |
Ref | Expression |
---|---|
ax11i.1 | ⊢ (x = y → (φ ↔ ψ)) |
ax11i.2 | ⊢ (ψ → ∀xψ) |
Ref | Expression |
---|---|
ax11i | ⊢ (x = y → (φ → ∀x(x = y → φ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax11i.1 | . 2 ⊢ (x = y → (φ ↔ ψ)) | |
2 | ax11i.2 | . . 3 ⊢ (ψ → ∀xψ) | |
3 | 1 | biimprcd 216 | . . 3 ⊢ (ψ → (x = y → φ)) |
4 | 2, 3 | alrimih 1565 | . 2 ⊢ (ψ → ∀x(x = y → φ)) |
5 | 1, 4 | syl6bi 219 | 1 ⊢ (x = y → (φ → ∀x(x = y → φ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∀wal 1540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 |
This theorem depends on definitions: df-bi 177 |
This theorem is referenced by: ax11wlem 1720 |
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