Theorem ax10lem5 1942

Description: Lemma for ax10 1944. Change free and bound variables. (Contributed by NM, 22-Jul-2015.)

Assertion

Ref	Expression
ax10lem5	⊢ (∀z z = w → ∀y y = x)

Distinct variable group:   z,w

Proof of Theorem ax10lem5

Dummy variables v u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax10lem1 1936	. . . 4 ⊢ (∀z z = w → ∀v v = w)
2		ax10lem4 1941	. . . 4 ⊢ (∀v v = w → ∀u u = v)
3	1, 2	syl 15	. . 3 ⊢ (∀z z = w → ∀u u = v)
4		ax10lem1 1936	. . 3 ⊢ (∀u u = v → ∀x x = v)
5	3, 4	syl 15	. 2 ⊢ (∀z z = w → ∀x x = v)
6		ax10lem4 1941	. 2 ⊢ (∀x x = v → ∀y y = x)
7	5, 6	syl 15	1 ⊢ (∀z z = w → ∀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-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  ax10  1944  a16g  1945  aev  1991