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Mirrors > Home > MPE Home > Th. List > ax7 | Structured version Visualization version GIF version |
Description: Proof of ax-7 2006
from ax7v1 2008 and ax7v2 2009 (and earlier axioms), proving
sufficiency of the conjunction of the latter two weakened versions of
ax7v 2007, which is itself a weakened version of ax-7 2006.
Note that the weakened version of ax-7 2006 obtained by adding a disjoint variable condition on 𝑥, 𝑧 (resp. on 𝑦, 𝑧) does not permit, together with the other axioms, to prove reflexivity (resp. symmetry). (Contributed by BJ, 7-Dec-2020.) |
Ref | Expression |
---|---|
ax7 | ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax7v2 2009 | . . . 4 ⊢ (𝑥 = 𝑡 → (𝑥 = 𝑦 → 𝑡 = 𝑦)) | |
2 | ax7v2 2009 | . . . 4 ⊢ (𝑥 = 𝑡 → (𝑥 = 𝑧 → 𝑡 = 𝑧)) | |
3 | ax7v1 2008 | . . . . . 6 ⊢ (𝑡 = 𝑦 → (𝑡 = 𝑧 → 𝑦 = 𝑧)) | |
4 | 3 | imp 407 | . . . . 5 ⊢ ((𝑡 = 𝑦 ∧ 𝑡 = 𝑧) → 𝑦 = 𝑧) |
5 | 4 | a1i 11 | . . . 4 ⊢ (𝑥 = 𝑡 → ((𝑡 = 𝑦 ∧ 𝑡 = 𝑧) → 𝑦 = 𝑧)) |
6 | 1, 2, 5 | syl2and 607 | . . 3 ⊢ (𝑥 = 𝑡 → ((𝑥 = 𝑦 ∧ 𝑥 = 𝑧) → 𝑦 = 𝑧)) |
7 | ax6evr 2013 | . . 3 ⊢ ∃𝑡 𝑥 = 𝑡 | |
8 | 6, 7 | exlimiiv 1923 | . 2 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 = 𝑧) → 𝑦 = 𝑧) |
9 | 8 | ex 413 | 1 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 |
This theorem is referenced by: equcomi 2015 equtr 2019 equequ1 2023 cbvaev 2049 aeveq 2052 axc16i 2450 equvel 2471 mo4 2643 axextnd 10001 bj-dtru 34036 wl-aetr 34651 wl-exeq 34655 wl-aleq 34656 wl-nfeqfb 34657 equcomi1 35916 hbequid 35925 equidqe 35938 aev-o 35947 ax6e2eq 40768 ax6e2eqVD 41118 2reu8i 43189 |
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