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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 2176. Same as Axiom C8 of [Monk2] p. 105. Use ax12v 2176 instead when sufficient. (Contributed by NM, 5-Aug-1993.) Remove dependencies on ax-10 2141 and ax-13 2371. (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 2030 | . . 3 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑦 → 𝑥 = 𝑧)) | |
2 | ax12v 2176 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) | |
3 | 1 | imim1d 82 | . . . . 5 ⊢ (𝑦 = 𝑧 → ((𝑥 = 𝑧 → 𝜑) → (𝑥 = 𝑦 → 𝜑))) |
4 | 3 | alimdv 1924 | . . . 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 2023 | . 2 ⊢ ∃𝑧 𝑦 = 𝑧 | |
8 | 6, 7 | exlimiiv 1939 | 1 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 |
This theorem is referenced by: sbalex 2240 equs5av 2275 wl-lem-exsb 35458 wl-lem-moexsb 35460 |
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