Theorem ceqex 2970

Description: Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.)

Assertion

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

Distinct variable group:   x,A

Allowed substitution hint:   φ(x)

Proof of Theorem ceqex

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		19.8a 1756	. . 3 ⊢ (x = A → ∃x x = A)
2		isset 2864	. . 3 ⊢ (A ∈ V ↔ ∃x x = A)
3	1, 2	sylibr 203	. 2 ⊢ (x = A → A ∈ V)
4		eqeq2 2362	. . . 4 ⊢ (y = A → (x = y ↔ x = A))
5	4	anbi1d 685	. . . . . 6 ⊢ (y = A → ((x = y ∧ φ) ↔ (x = A ∧ φ)))
6	5	exbidv 1626	. . . . 5 ⊢ (y = A → (∃x(x = y ∧ φ) ↔ ∃x(x = A ∧ φ)))
7	6	bibi2d 309	. . . 4 ⊢ (y = A → ((φ ↔ ∃x(x = y ∧ φ)) ↔ (φ ↔ ∃x(x = A ∧ φ))))
8	4, 7	imbi12d 311	. . 3 ⊢ (y = A → ((x = y → (φ ↔ ∃x(x = y ∧ φ))) ↔ (x = A → (φ ↔ ∃x(x = A ∧ φ)))))
9		19.8a 1756	. . . . 5 ⊢ ((x = y ∧ φ) → ∃x(x = y ∧ φ))
10	9	ex 423	. . . 4 ⊢ (x = y → (φ → ∃x(x = y ∧ φ)))
11		vex 2863	. . . . . 6 ⊢ y ∈ V
12	11	alexeq 2969	. . . . 5 ⊢ (∀x(x = y → φ) ↔ ∃x(x = y ∧ φ))
13		sp 1747	. . . . . 6 ⊢ (∀x(x = y → φ) → (x = y → φ))
14	13	com12 27	. . . . 5 ⊢ (x = y → (∀x(x = y → φ) → φ))
15	12, 14	syl5bir 209	. . . 4 ⊢ (x = y → (∃x(x = y ∧ φ) → φ))
16	10, 15	impbid 183	. . 3 ⊢ (x = y → (φ ↔ ∃x(x = y ∧ φ)))
17	8, 16	vtoclg 2915	. 2 ⊢ (A ∈ V → (x = A → (φ ↔ ∃x(x = A ∧ φ))))
18	3, 17	mpcom 32	1 ⊢ (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-or 359  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-nfc 2479  df-v 2862

This theorem is referenced by:  ceqsexg  2971  sbc6g  3072