Theorem rexsn 4642

Description: Convert an existential quantification restricted to a singleton to a substitution. (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 4636	. 2 ⊢ (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))
4	1, 3	ax-mp 5	1 ⊢ (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  Vcvv 3444  {csn 4585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-v 3446  df-sn 4586

This theorem is referenced by:  elsnres  5981  oarec  8503  snec  8728  zornn0g  10434  fpwwe2lem12  10571  elreal  11060  hashge2el2difr  14422  vdwlem6  16933  pzriprnglem10  21376  pmatcollpw3fi1  22651  restsn  23033  snclseqg  23979  ust0  24083  0slt1s  27717  cuteq1  27722  made0  27761  cofcutr  27808  mulsrid  27992  n0scut  28202  n0sfincut  28222  zscut  28271  1p1e2s  28278  twocut  28285  halfcut  28309  addhalfcut  28310  grplsm0l  33347  rprmdvdsprod  33478  esum2dlem  34055  eulerpartlemgh  34342  eldm3  35721  poimirlem28  37615  heiborlem3  37780  tfsconcatrn  43304  nregmodel  44980  stgr1  47933  setc1onsubc  49564