Theorem sbc6g 3072

Description: An equivalence for class substitution. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Assertion

Ref	Expression
sbc6g	⊢ (A ∈ V → ([̣A / x]̣φ ↔ ∀x(x = A → φ)))

Distinct variable group:   x,A

Allowed substitution hints:   φ(x)   V(x)

Proof of Theorem sbc6g

Step	Hyp	Ref	Expression
1		nfe1 1732	. . 3 ⊢ Ⅎx∃x(x = A ∧ φ)
2		ceqex 2970	. . 3 ⊢ (x = A → (φ ↔ ∃x(x = A ∧ φ)))
3	1, 2	ceqsalg 2884	. 2 ⊢ (A ∈ V → (∀x(x = A → φ) ↔ ∃x(x = A ∧ φ)))
4		sbc5 3071	. 2 ⊢ ([̣A / x]̣φ ↔ ∃x(x = A ∧ φ))
5	3, 4	syl6rbbr 255	1 ⊢ (A ∈ V → ([̣A / x]̣φ ↔ ∀x(x = A → φ)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  [̣wsbc 3047

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  df-sbc 3048

This theorem is referenced by:  sbc6  3073  sbciegft  3077  ralsns  3764