Theorem ceqsexgv 2971

Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 29-Dec-1996.)

Hypothesis

Ref	Expression
ceqsexgv.1	⊢ (x = A → (φ ↔ ψ))

Assertion

Ref	Expression
ceqsexgv	⊢ (A ∈ V → (∃x(x = A ∧ φ) ↔ ψ))

Distinct variable groups:   x,A   ψ,x

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

Proof of Theorem ceqsexgv

Step	Hyp	Ref	Expression
1		nfv 1619	. 2 ⊢ Ⅎxψ
2		ceqsexgv.1	. 2 ⊢ (x = A → (φ ↔ ψ))
3	1, 2	ceqsexg 2970	1 ⊢ (A ∈ V → (∃x(x = A ∧ φ) ↔ ψ))

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

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 2478  df-v 2861

This theorem is referenced by:  ceqsrexv  2972  clel3g  2976  eluni1g  4172  opkelopkabg  4245  otkelins2kg  4253  otkelins3kg  4254  opkelcokg  4261