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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 34493 without using ax-c15 34493. Quantification case. When 𝑧 and 𝑦 are distinct, this theorem avoids the dummy variables needed by the more general ax12inda 34552. (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 34538 | . . . . 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 34549 | . 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))))) |
8 | 5, 7 | pm2.61i 176 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-c5 34487 ax-c4 34488 ax-c7 34489 ax-c10 34490 ax-c11 34491 ax-c9 34494 ax-c16 34496 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1526 df-ex 1745 df-nf 1750 |
This theorem is referenced by: ax12inda 34552 |
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