Theorem ceqsexv 2799

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 1539	. 2 ⊢ Ⅎ𝑥𝜓
2		ceqsexv.1	. 2 ⊢ 𝐴 ∈ V
3		ceqsexv.2	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	1, 2, 3	ceqsex 2798	1 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364  ∃wex 1503   ∈ wcel 2164  Vcvv 2760

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-v 2762

This theorem is referenced by:  ceqsex3v  2802  gencbvex  2806  sbhypf  2809  euxfr2dc  2945  inuni  4184  eqvinop  4272  onm  4432  uniuni  4482  opeliunxp  4714  elvvv  4722  rexiunxp  4804  imai  5021  coi1  5181  abrexco  5802  opabex3d  6173  opabex3  6174  mapsnen  6865  xpsnen  6875  xpcomco  6880  xpassen  6884