Theorem ax7 2039

Description: Proof of ax-7 2031 from ax7v1 2033 and ax7v2 2034 (and earlier axioms), proving sufficiency of the conjunction of the latter two weakened versions of ax7v 2032, which is itself a weakened version of ax-7 2031.

Note that the weakened version of ax-7 2031 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.)

Assertion

Ref	Expression
ax7	⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))

Proof of Theorem ax7

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax7v2 2034	. . . 4 ⊢ (𝑥 = 𝑡 → (𝑥 = 𝑦 → 𝑡 = 𝑦))
2		ax7v2 2034	. . . 4 ⊢ (𝑥 = 𝑡 → (𝑥 = 𝑧 → 𝑡 = 𝑧))
3		ax7v1 2033	. . . . . 6 ⊢ (𝑡 = 𝑦 → (𝑡 = 𝑧 → 𝑦 = 𝑧))
4	3	imp 411	. . . . 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 619	. . 3 ⊢ (𝑥 = 𝑡 → ((𝑥 = 𝑦 ∧ 𝑥 = 𝑧) → 𝑦 = 𝑧))
7		ax6evr 2038	. . 3 ⊢ ∃𝑡 𝑥 = 𝑡
8	6, 7	exlimiiv 1954	. 2 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 = 𝑧) → 𝑦 = 𝑧)
9	8	ex 417	1 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))

Syntax hints:   → wi 4   ∧ wa 400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1803

This theorem is referenced by:  equcomi  2040  equtr  2044  equequ1  2048  cbvaev  2078  aeveq  2081  axc16i  2470  equvel  2490  mo4  2596  axextnd  10564  in-ax8  36592  ss-ax8  36593  wl-aetr  38039  wl-exeq  38044  wl-aleq  38045  wl-nfeqfb  38046  equcomi1  39531  hbequid  39540  equidqe  39553  aev-o  39562  ax6e2eq  45125  ax6e2eqVD  45474  et-equeucl  47445  2reu8i  47706