Theorem equs5a 1887

Description: A property related to substitution that unlike equs5 1996 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)

Assertion

Ref	Expression
equs5a	⊢ (∃x(x = y ∧ ∀yφ) → ∀x(x = y → φ))

Proof of Theorem equs5a

Step	Hyp	Ref	Expression
1		nfa1 1788	. 2 ⊢ Ⅎx∀x(x = y → φ)
2		ax-11 1746	. . 3 ⊢ (x = y → (∀yφ → ∀x(x = y → φ)))
3	2	imp 418	. 2 ⊢ ((x = y ∧ ∀yφ) → ∀x(x = y → φ))
4	1, 3	exlimi 1803	1 ⊢ (∃x(x = y ∧ ∀yφ) → ∀x(x = y → φ))

Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541

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:  sb4a  1923  equs45f  1989