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Mirrors > Home > MPE Home > Th. List > Mathboxes > ax12inda2 | Structured version Visualization version GIF version |
Description: Induction step for constructing a substitution instance of ax-c15 35905 without using ax-c15 35905. Quantification case. When 𝑧 and 𝑦 are distinct, this theorem avoids the dummy variables needed by the more general ax12inda 35964. (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ax12inda2.1 | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) |
Ref | Expression |
---|---|
ax12inda2 | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1 6 | . . . . 5 ⊢ (∀𝑧𝜑 → (𝑥 = 𝑦 → ∀𝑧𝜑)) | |
2 | axc16g-o 35950 | . . . . 5 ⊢ (∀𝑦 𝑦 = 𝑧 → ((𝑥 = 𝑦 → ∀𝑧𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))) | |
3 | 1, 2 | syl5 34 | . . . 4 ⊢ (∀𝑦 𝑦 = 𝑧 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))) |
4 | 3 | a1d 25 | . . 3 ⊢ (∀𝑦 𝑦 = 𝑧 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑)))) |
5 | 4 | a1d 25 | . 2 ⊢ (∀𝑦 𝑦 = 𝑧 → (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))))) |
6 | ax12inda2.1 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) | |
7 | 6 | ax12indalem 35961 | . 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))))) |
8 | 5, 7 | pm2.61i 183 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1526 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-11 2151 ax-12 2167 ax-13 2381 ax-c5 35899 ax-c4 35900 ax-c7 35901 ax-c10 35902 ax-c11 35903 ax-c9 35906 ax-c16 35908 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 |
This theorem is referenced by: ax12inda 35964 |
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