Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > ax12a2-o | Structured version Visualization version GIF version |
Description: Derive ax-c15 33192 from a hypothesis in the form of ax-12 2034, without using ax-12 2034 or ax-c15 33192. The hypothesis is weaker than ax-12 2034, with 𝑧 both distinct from 𝑥 and not occurring in 𝜑. Thus, the hypothesis provides an alternate axiom that can be used in place of ax-12 2034, if we also have ax-c11 33190, which this proof uses. As theorem ax12 2292 shows, the distinct variable conditions are optional. An open problem is whether we can derive this with ax-c11n 33191 instead of ax-c11 33190. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ax12a2-o.1 | ⊢ (𝑥 = 𝑧 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) |
Ref | Expression |
---|---|
ax12a2-o | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-5 1827 | . . 3 ⊢ (𝜑 → ∀𝑧𝜑) | |
2 | ax12a2-o.1 | . . 3 ⊢ (𝑥 = 𝑧 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) | |
3 | 1, 2 | syl5 33 | . 2 ⊢ (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) |
4 | 3 | ax12v2-o 33252 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-c5 33186 ax-c4 33187 ax-c7 33188 ax-c10 33189 ax-c11 33190 ax-c9 33193 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |