Theorem rexsn 4641

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 4635	. 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 1542   ∈ wcel 2114  ∃wrex 3062  Vcvv 3442  {csn 4582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-v 3444  df-sn 4583

This theorem is referenced by:  elsnres  5988  oarec  8499  snec  8727  zornn0g  10427  fpwwe2lem12  10565  elreal  11054  hashge2el2difr  14416  vdwlem6  16926  pzriprnglem10  21457  pmatcollpw3fi1  22744  restsn  23126  snclseqg  24072  ust0  24176  0lt1s  27820  cuteq1  27825  made0  27871  cofcutr  27932  mulsrid  28121  n0cut  28342  n0fincut  28363  zcuts  28415  twocut  28431  halfcut  28466  addhalfcut  28467  pw2cut2  28470  domnprodeq0  33370  grplsm0l  33496  rprmdvdsprod  33627  esum2dlem  34270  eulerpartlemgh  34556  eldm3  35977  poimirlem28  37899  heiborlem3  38064  tfsconcatrn  43699  nregmodel  45373  stgr1  48321  setc1onsubc  49961