Theorem rexsn 3568

Description: Restricted existential quantification over a singleton. (Contributed by Jeff Madsen, 5-Jan-2011.)

Hypotheses

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

Assertion

Ref	Expression
rexsn	⊢ (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓)

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rexsn

Step	Hyp	Ref	Expression
1		ralsn.1	. 2 ⊢ 𝐴 ∈ V
2		ralsn.2	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3	2	rexsng 3565	. 2 ⊢ (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))
4	1, 3	ax-mp 5	1 ⊢ (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1331   ∈ wcel 1480  ∃wrex 2417  Vcvv 2686  {csn 3527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-sbc 2910  df-sn 3533

This theorem is referenced by:  elsnres  4856  snec  6490  0ct  6992  elreal  7636  restsn  12349