Theorem equs4 1959

Description: Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.)

Assertion

Ref	Expression
equs4	⊢ (∀x(x = y → φ) → ∃x(x = y ∧ φ))

Proof of Theorem equs4

Step	Hyp	Ref	Expression
1		a9e 1951	. . 3 ⊢ ∃x x = y
2		19.29 1596	. . 3 ⊢ ((∀x(x = y → φ) ∧ ∃x x = y) → ∃x((x = y → φ) ∧ x = y))
3	1, 2	mpan2 652	. 2 ⊢ (∀x(x = y → φ) → ∃x((x = y → φ) ∧ x = y))
4		ancl 529	. . . 4 ⊢ ((x = y → φ) → (x = y → (x = y ∧ φ)))
5	4	imp 418	. . 3 ⊢ (((x = y → φ) ∧ x = y) → (x = y ∧ φ))
6	5	eximi 1576	. 2 ⊢ (∃x((x = y → φ) ∧ x = y) → ∃x(x = y ∧ φ))
7	3, 6	syl 15	1 ⊢ (∀x(x = y → φ) → ∃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-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:  equs45f  1989  sb2  2023