Theorem ax10lem1 1936

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

Assertion

Ref	Expression
ax10lem1	⊢ (∀x x = w → ∀y y = w)

Distinct variable groups:   x,w   y,w

Proof of Theorem ax10lem1

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-8 1675	. . 3 ⊢ (x = v → (x = w → v = w))
2	1	cbvalivw 1674	. 2 ⊢ (∀x x = w → ∀v v = w)
3		ax-8 1675	. . 3 ⊢ (v = y → (v = w → y = w))
4	3	cbvalivw 1674	. 2 ⊢ (∀v v = w → ∀y y = w)
5	2, 4	syl 15	1 ⊢ (∀x x = w → ∀y y = w)

This proof depends on syntax axioms:   → wi 4  ∀wal 1540

This proof depends on 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

This proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is used by:  ax10lem3  1938  ax10lem4  1941  ax10lem5  1942