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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ax12a2-o | Structured version Visualization version GIF version |
Description: Derive ax-c15 36185 from a hypothesis in the form of ax-12 2175, without using ax-12 2175 or ax-c15 36185. The hypothesis is weaker than ax-12 2175, 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 2175, if we also have ax-c11 36183, which this proof uses. As theorem ax12 2434 shows, the distinct variable conditions are optional. An open problem is whether we can derive this with ax-c11n 36184 instead of ax-c11 36183. (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 1911 | . . 3 ⊢ (𝜑 → ∀𝑧𝜑) | |
2 | ax12a2-o.1 | . . 3 ⊢ (𝑥 = 𝑧 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) | |
3 | 1, 2 | syl5 34 | . 2 ⊢ (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) |
4 | 3 | ax12v2-o 36245 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2142 ax-11 2158 ax-12 2175 ax-13 2379 ax-c5 36179 ax-c4 36180 ax-c7 36181 ax-c10 36182 ax-c11 36183 ax-c9 36186 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 |
This theorem is referenced by: (None) |
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