Theorem ax10lem3 1938

Description: Lemma for ax10 1944. Similar to ax-10 2140 but with distinct variables. (Contributed by NM, 25-Jul-2015.)

Assertion

Ref	Expression
ax10lem3	⊢ (∀x x = y → ∀y y = x)

Distinct variable group:   x,y

Proof of Theorem ax10lem3

Dummy variables w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax10lem2 1937	. 2 ⊢ (∀x x = y → ∀x x = z)
2		ax10lem1 1936	. . . 4 ⊢ (∀x x = z → ∀w w = z)
3		ax10lem2 1937	. . . 4 ⊢ (∀w w = z → ∀w w = x)
4	2, 3	syl 15	. . 3 ⊢ (∀x x = z → ∀w w = x)
5		ax10lem1 1936	. . 3 ⊢ (∀w w = x → ∀y y = x)
6	4, 5	syl 15	. 2 ⊢ (∀x x = z → ∀y y = x)
7	1, 6	syl 15	1 ⊢ (∀x x = y → ∀y y = x)

Syntax hints:   → wi 4  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545

This theorem is referenced by:  dvelimv  1939