Theorem ax7 2013

Description: Proof of ax-7 2005 from ax7v1 2007 and ax7v2 2008 (and earlier axioms), proving sufficiency of the conjunction of the latter two weakened versions of ax7v 2006, which is itself a weakened version of ax-7 2005.

Note that the weakened version of ax-7 2005 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 2008	. . . 4 ⊢ (𝑥 = 𝑡 → (𝑥 = 𝑦 → 𝑡 = 𝑦))
2		ax7v2 2008	. . . 4 ⊢ (𝑥 = 𝑡 → (𝑥 = 𝑧 → 𝑡 = 𝑧))
3		ax7v1 2007	. . . . . 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 2012	. . 3 ⊢ ∃𝑡 𝑥 = 𝑡
8	6, 7	exlimiiv 1929	. 2 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 = 𝑧) → 𝑦 = 𝑧)
9	8	ex 412	1 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))

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 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777

This theorem is referenced by:  equcomi  2014  equtr  2018  equequ1  2022  cbvaev  2051  aeveq  2054  axc16i  2439  equvel  2459  mo4  2564  axextnd  10629  in-ax8  36207  ss-ax8  36208  wl-aetr  37510  wl-exeq  37515  wl-aleq  37516  wl-nfeqfb  37517  equcomi1  38882  hbequid  38891  equidqe  38904  aev-o  38913  ax6e2eq  44555  ax6e2eqVD  44905  et-equeucl  46828  2reu8i  47063