Theorem equs5 1996

Description: Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
equs5	⊢ (¬ ∀x x = y → (∃x(x = y ∧ φ) → ∀x(x = y → φ)))

Proof of Theorem equs5

Step	Hyp	Ref	Expression
1		nfnae 1956	. 2 ⊢ Ⅎx ¬ ∀x x = y
2		nfa1 1788	. 2 ⊢ Ⅎx∀x(x = y → φ)
3		ax11o 1994	. . 3 ⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ))))
4	3	imp3a 420	. 2 ⊢ (¬ ∀x x = y → ((x = y ∧ φ) → ∀x(x = y → φ)))
5	1, 2, 4	exlimd 1806	1 ⊢ (¬ ∀x x = y → (∃x(x = y ∧ φ) → ∀x(x = y → φ)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541

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-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

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

This theorem is used by:  sb3  2052  sb4  2053