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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 34696 from ax-c11n 34695 and the shorter
ax-12 2203 that has replaced it.
Theorem ax12 2460 shows the reverse derivation of ax-12 2203 from ax-c15 34696. Normally, axc15 2459 should be used rather than ax-c15 34696, except by theorems specifically studying the latter's properties. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 21-Apr-2018.) |
Ref | Expression |
---|---|
axc15 | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax6ev 2059 | . 2 ⊢ ∃𝑧 𝑧 = 𝑦 | |
2 | dveeq2 2454 | . . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) | |
3 | ax12v 2204 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) | |
4 | equequ2 2111 | . . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦)) | |
5 | 4 | sps 2209 | . . . . . 6 ⊢ (∀𝑥 𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦)) |
6 | nfa1 2184 | . . . . . . . 8 ⊢ Ⅎ𝑥∀𝑥 𝑧 = 𝑦 | |
7 | 5 | imbi1d 330 | . . . . . . . 8 ⊢ (∀𝑥 𝑧 = 𝑦 → ((𝑥 = 𝑧 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑))) |
8 | 6, 7 | albid 2246 | . . . . . . 7 ⊢ (∀𝑥 𝑧 = 𝑦 → (∀𝑥(𝑥 = 𝑧 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
9 | 8 | imbi2d 329 | . . . . . 6 ⊢ (∀𝑥 𝑧 = 𝑦 → ((𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) |
10 | 5, 9 | imbi12d 333 | . . . . 5 ⊢ (∀𝑥 𝑧 = 𝑦 → ((𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) ↔ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))) |
11 | 3, 10 | mpbii 223 | . . . 4 ⊢ (∀𝑥 𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) |
12 | 2, 11 | syl6 35 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))) |
13 | 12 | exlimdv 2013 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))) |
14 | 1, 13 | 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 ↔ wb 196 ∀wal 1629 ∃wex 1852 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-10 2174 ax-12 2203 ax-13 2408 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-ex 1853 df-nf 1858 |
This theorem is referenced by: ax12 2460 ax12b 2491 equs5 2497 ax12vALT 2575 bj-ax12v3ALT 33012 |
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