Theorem ax7 1930

Description: Proof of ax-7 1922 from ax7v1 1924 and ax7v2 1925, proving sufficiency of the conjunction of the latter two weakened versions of ax7v 1923, which is itself a weakened version of ax-7 1922.

Note that the weakened version of ax-7 1922 obtained by adding a dv 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 1925	. . . 4 ⊢ (𝑥 = 𝑡 → (𝑥 = 𝑦 → 𝑡 = 𝑦))
2		ax7v2 1925	. . . 4 ⊢ (𝑥 = 𝑡 → (𝑥 = 𝑧 → 𝑡 = 𝑧))
3		ax7v1 1924	. . . . . 6 ⊢ (𝑡 = 𝑦 → (𝑡 = 𝑧 → 𝑦 = 𝑧))
4	3	imp 444	. . . . 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 499	. . 3 ⊢ (𝑥 = 𝑡 → ((𝑥 = 𝑦 ∧ 𝑥 = 𝑧) → 𝑦 = 𝑧))
7	6	expd 451	. 2 ⊢ (𝑥 = 𝑡 → (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧)))
8		ax6evr 1929	. 2 ⊢ ∃𝑡 𝑥 = 𝑡
9	7, 8	exlimiiv 1846	1 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))

Syntax hints:   → wi 4   ∧ wa 383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696

This theorem is referenced by:  equcomi  1931  equtr  1935  equequ1  1939  equvinv  1946  cbvaev  1966  aeveq  1969  aevOLD  2148  aevALTOLD  2309  axc16i  2310  equvel  2335  axext3  2592  dtru  4783  axextnd  9292  2spotmdisj  26595  bj-dtru  31985  bj-mo3OLD  32022  wl-aetr  32496  wl-exeq  32500  wl-aleq  32501  wl-nfeqfb  32502  equcomi1  33203  hbequid  33212  equidqe  33225  aev-o  33234  ax6e2eq  37794  ax6e2eqVD  38165