Theorem ax7 2001

Description: Proof of ax-7 1993 from ax7v1 1995 and ax7v2 1996 (and earlier axioms), proving sufficiency of the conjunction of the latter two weakened versions of ax7v 1994, which is itself a weakened version of ax-7 1993.

Note that the weakened version of ax-7 1993 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 1996	. . . 4 ⊢ (𝑥 = 𝑡 → (𝑥 = 𝑦 → 𝑡 = 𝑦))
2		ax7v2 1996	. . . 4 ⊢ (𝑥 = 𝑡 → (𝑥 = 𝑧 → 𝑡 = 𝑧))
3		ax7v1 1995	. . . . . 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 2000	. . 3 ⊢ ∃𝑡 𝑥 = 𝑡
8	6, 7	exlimiiv 1910	. 2 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 = 𝑧) → 𝑦 = 𝑧)
9	8	ex 413	1 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))

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 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1763

This theorem is referenced by:  equcomi  2002  equtr  2006  equequ1  2010  cbvaev  2030  aeveq  2033  axc16i  2414  equvel  2435  mo4  2606  axextnd  9862  bj-dtru  33702  wl-aetr  34315  wl-exeq  34319  wl-aleq  34320  wl-nfeqfb  34321  equcomi1  35580  hbequid  35589  equidqe  35602  aev-o  35611  ax6e2eq  40443  ax6e2eqVD  40793  2reu8i  42842