Theorem ax7 2015

Description: Proof of ax-7 2007 from ax7v1 2009 and ax7v2 2010 (and earlier axioms), proving sufficiency of the conjunction of the latter two weakened versions of ax7v 2008, which is itself a weakened version of ax-7 2007.

Note that the weakened version of ax-7 2007 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 2010	. . . 4 ⊢ (𝑥 = 𝑡 → (𝑥 = 𝑦 → 𝑡 = 𝑦))
2		ax7v2 2010	. . . 4 ⊢ (𝑥 = 𝑡 → (𝑥 = 𝑧 → 𝑡 = 𝑧))
3		ax7v1 2009	. . . . . 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 607	. . 3 ⊢ (𝑥 = 𝑡 → ((𝑥 = 𝑦 ∧ 𝑥 = 𝑧) → 𝑦 = 𝑧))
7		ax6evr 2014	. . 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))

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 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778

This theorem is referenced by:  equcomi  2016  equtr  2020  equequ1  2024  cbvaev  2053  aeveq  2056  axc16i  2444  equvel  2464  mo4  2569  axextnd  10660  in-ax8  36190  ss-ax8  36191  wl-aetr  37483  wl-exeq  37488  wl-aleq  37489  wl-nfeqfb  37490  equcomi1  38856  hbequid  38865  equidqe  38878  aev-o  38887  ax6e2eq  44528  ax6e2eqVD  44878  et-equeucl  46793  2reu8i  47028