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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 36027 from ax-c11n 36026 and the shorter
ax-12 2177 that has replaced it.
Theorem ax12 2445 shows the reverse derivation of ax-12 2177 from ax-c15 36027. Normally, axc15 2444 should be used rather than ax-c15 36027, except by theorems specifically studying the latter's properties. Usage of this theorem is discouraged because it depends on ax-13 2390. (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 1972 | . 2 ⊢ ∃𝑧 𝑧 = 𝑦 | |
2 | dveeq2 2396 | . . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) | |
3 | ax12v 2178 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) | |
4 | equeuclr 2030 | . . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑦 → 𝑥 = 𝑧)) | |
5 | 4 | sps 2184 | . . . . 5 ⊢ (∀𝑥 𝑧 = 𝑦 → (𝑥 = 𝑦 → 𝑥 = 𝑧)) |
6 | 4 | imim1d 82 | . . . . . . 7 ⊢ (𝑧 = 𝑦 → ((𝑥 = 𝑧 → 𝜑) → (𝑥 = 𝑦 → 𝜑))) |
7 | 6 | al2imi 1816 | . . . . . 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 1934 | . 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 1535 ∃wex 1780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-12 2177 ax-13 2390 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-nf 1785 |
This theorem is referenced by: ax12 2445 ax12b 2446 equs5 2483 ax12vALT 2492 bj-ax12v3ALT 34022 |
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