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Mirrors > Home > MPE Home > Th. List > axc15 | Structured version Visualization version GIF version |
Description: Derivation of set.mm's
original ax-c15 36903 from ax-c11n 36902 and the shorter
ax-12 2171 that has replaced it.
Theorem ax12 2423 shows the reverse derivation of ax-12 2171 from ax-c15 36903. Normally, axc15 2422 should be used rather than ax-c15 36903, except by theorems specifically studying the latter's properties. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 26-Mar-2023.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axc15 | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax6ev 1973 | . 2 ⊢ ∃𝑧 𝑧 = 𝑦 | |
2 | dveeq2 2378 | . . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) | |
3 | ax12v 2172 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) | |
4 | equeuclr 2026 | . . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑦 → 𝑥 = 𝑧)) | |
5 | 4 | sps 2178 | . . . . 5 ⊢ (∀𝑥 𝑧 = 𝑦 → (𝑥 = 𝑦 → 𝑥 = 𝑧)) |
6 | 4 | imim1d 82 | . . . . . . 7 ⊢ (𝑧 = 𝑦 → ((𝑥 = 𝑧 → 𝜑) → (𝑥 = 𝑦 → 𝜑))) |
7 | 6 | al2imi 1818 | . . . . . 6 ⊢ (∀𝑥 𝑧 = 𝑦 → (∀𝑥(𝑥 = 𝑧 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
8 | 7 | imim2d 57 | . . . . 5 ⊢ (∀𝑥 𝑧 = 𝑦 → ((𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)) → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) |
9 | 5, 8 | imim12d 81 | . . . 4 ⊢ (∀𝑥 𝑧 = 𝑦 → ((𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))) |
10 | 2, 3, 9 | syl6mpi 67 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))) |
11 | 10 | exlimdv 1936 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))) |
12 | 1, 11 | mpi 20 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1537 ∃wex 1782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-10 2137 ax-12 2171 ax-13 2372 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1783 df-nf 1787 |
This theorem is referenced by: ax12 2423 ax12b 2424 equs5 2460 ax12vALT 2469 bj-ax12v3ALT 34868 |
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