Theorem ceqsexv 2802

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 1542	. 2 ⊢ Ⅎ𝑥𝜓
2		ceqsexv.1	. 2 ⊢ 𝐴 ∈ V
3		ceqsexv.2	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	1, 2, 3	ceqsex 2801	1 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364  ∃wex 1506   ∈ wcel 2167  Vcvv 2763

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-v 2765

This theorem is referenced by:  ceqsex3v  2806  gencbvex  2810  sbhypf  2813  euxfr2dc  2949  inuni  4188  eqvinop  4276  onm  4436  uniuni  4486  opeliunxp  4718  elvvv  4726  rexiunxp  4808  imai  5025  coi1  5185  abrexco  5806  opabex3d  6178  opabex3  6179  mapsnen  6870  xpsnen  6880  xpcomco  6885  xpassen  6889