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| 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 406 | . . . . 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 608 | . . 3 ⊢ (𝑥 = 𝑡 → ((𝑥 = 𝑦 ∧ 𝑥 = 𝑧) → 𝑦 = 𝑧)) | 
| 7 | ax6evr 2013 | . . 3 ⊢ ∃𝑡 𝑥 = 𝑡 | |
| 8 | 6, 7 | exlimiiv 1930 | . 2 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 = 𝑧) → 𝑦 = 𝑧) | 
| 9 | 8 | ex 412 | 1 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 | 
| This theorem is referenced by: equcomi 2015 equtr 2019 equequ1 2023 cbvaev 2052 aeveq 2055 axc16i 2440 equvel 2460 mo4 2565 axextnd 10632 in-ax8 36226 ss-ax8 36227 wl-aetr 37531 wl-exeq 37536 wl-aleq 37537 wl-nfeqfb 37538 equcomi1 38902 hbequid 38911 equidqe 38924 aev-o 38933 ax6e2eq 44582 ax6e2eqVD 44932 et-equeucl 46892 2reu8i 47130 | 
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