Theorem alexeq 2969

Description: Two ways to express substitution of A for x in φ. (Contributed by NM, 2-Mar-1995.)

Hypothesis

Ref	Expression
alexeq.1	⊢ A ∈ V

Assertion

Ref	Expression
alexeq	⊢ (∀x(x = A → φ) ↔ ∃x(x = A ∧ φ))

Distinct variable group:   x,A

Allowed substitution hint:   φ(x)

Proof of Theorem alexeq

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		alexeq.1	. . 3 ⊢ A ∈ V
2		eqeq2 2362	. . . . 5 ⊢ (y = A → (x = y ↔ x = A))
3	2	anbi1d 685	. . . 4 ⊢ (y = A → ((x = y ∧ φ) ↔ (x = A ∧ φ)))
4	3	exbidv 1626	. . 3 ⊢ (y = A → (∃x(x = y ∧ φ) ↔ ∃x(x = A ∧ φ)))
5	2	imbi1d 308	. . . 4 ⊢ (y = A → ((x = y → φ) ↔ (x = A → φ)))
6	5	albidv 1625	. . 3 ⊢ (y = A → (∀x(x = y → φ) ↔ ∀x(x = A → φ)))
7		sb56 2098	. . 3 ⊢ (∃x(x = y ∧ φ) ↔ ∀x(x = y → φ))
8	1, 4, 6, 7	vtoclb 2913	. 2 ⊢ (∃x(x = A ∧ φ) ↔ ∀x(x = A → φ))
9	8	bicomi 193	1 ⊢ (∀x(x = A → φ) ↔ ∃x(x = A ∧ φ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860

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  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862

This theorem is referenced by:  ceqex  2970