Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ax12v2 | Structured version Visualization version GIF version |
Description: It is possible to remove any restriction on 𝜑 in ax12v 2048. Same as Axiom C8 of [Monk2] p. 105. Use ax12v 2048 instead when sufficient. (Contributed by NM, 5-Aug-1993.) Removed dependencies on ax-10 2019 and ax-13 2246. (Revised by Jim Kingdon, 15-Dec-2017.) (Proof shortened by Wolf Lammen, 8-Dec-2019.) |
Ref | Expression |
---|---|
ax12v2 | ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equtrr 1949 | . . 3 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑦 → 𝑥 = 𝑧)) | |
2 | ax12v 2048 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) | |
3 | 1 | imim1d 82 | . . . . 5 ⊢ (𝑦 = 𝑧 → ((𝑥 = 𝑧 → 𝜑) → (𝑥 = 𝑦 → 𝜑))) |
4 | 3 | alimdv 1845 | . . . 4 ⊢ (𝑦 = 𝑧 → (∀𝑥(𝑥 = 𝑧 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
5 | 2, 4 | syl9r 78 | . . 3 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) |
6 | 1, 5 | syld 47 | . 2 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) |
7 | ax6evr 1942 | . 2 ⊢ ∃𝑧 𝑦 = 𝑧 | |
8 | 6, 7 | exlimiiv 1859 | 1 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-12 2047 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 |
This theorem is referenced by: axc11rvOLD 2140 sb56 2150 bj-ax12 32634 wl-lem-exsb 33348 wl-lem-moexsb 33350 |
Copyright terms: Public domain | W3C validator |