Theorem ceqsexv 2811

Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.)

Hypotheses

Ref	Expression
ceqsexv.1	⊢ 𝐴 ∈ V
ceqsexv.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
ceqsexv	⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ceqsexv

Step	Hyp	Ref	Expression
1		nfv 1551	. 2 ⊢ Ⅎ𝑥𝜓
2		ceqsexv.1	. 2 ⊢ 𝐴 ∈ V
3		ceqsexv.2	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	1, 2, 3	ceqsex 2810	1 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1373  ∃wex 1515   ∈ wcel 2176  Vcvv 2772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-v 2774

This theorem is referenced by:  ceqsex3v  2815  gencbvex  2819  sbhypf  2822  euxfr2dc  2958  inuni  4200  eqvinop  4288  onm  4449  uniuni  4499  opeliunxp  4731  elvvv  4739  rexiunxp  4821  imai  5039  coi1  5199  abrexco  5830  opabex3d  6208  opabex3  6209  mapsnen  6905  xpsnen  6918  xpcomco  6923  xpassen  6927