Theorem equsexh 1963

Description: A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
equsexh.1	⊢ (ψ → ∀xψ)
equsexh.2	⊢ (x = y → (φ ↔ ψ))

Assertion

Ref	Expression
equsexh	⊢ (∃x(x = y ∧ φ) ↔ ψ)

Proof of Theorem equsexh

Step	Hyp	Ref	Expression
1		equsexh.1	. . 3 ⊢ (ψ → ∀xψ)
2	1	nfi 1551	. 2 ⊢ Ⅎxψ
3		equsexh.2	. 2 ⊢ (x = y → (φ ↔ ψ))
4	2, 3	equsex 1962	1 ⊢ (∃x(x = y ∧ φ) ↔ ψ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ 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:  cleljust  2014