Theorem ax10lem6 1943

Description: Lemma for ax10 1944. Similar to ax10o 1952 but with reversed antecedent. (Contributed by NM, 25-Jul-2015.)

Assertion

Ref	Expression
ax10lem6	⊢ (∀y y = x → (∀xφ → ∀yφ))

Proof of Theorem ax10lem6

Step	Hyp	Ref	Expression
1		ax-11 1746	. . 3 ⊢ (y = x → (∀xφ → ∀y(y = x → φ)))
2	1	sps 1754	. 2 ⊢ (∀y y = x → (∀xφ → ∀y(y = x → φ)))
3		pm2.27 35	. . 3 ⊢ (y = x → ((y = x → φ) → φ))
4	3	al2imi 1561	. 2 ⊢ (∀y y = x → (∀y(y = x → φ) → ∀yφ))
5	2, 4	syld 40	1 ⊢ (∀y y = x → (∀xφ → ∀yφ))

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  ax-11 1746

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

This theorem is used by:  ax10  1944