Theorem rexsn 4626

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 4620	. 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 3061  Vcvv 3429  {csn 4567

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 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-v 3431  df-sn 4568

This theorem is referenced by:  elsnres  5986  oarec  8497  snec  8725  zornn0g  10427  fpwwe2lem12  10565  elreal  11054  hashge2el2difr  14443  vdwlem6  16957  pzriprnglem10  21470  pmatcollpw3fi1  22753  restsn  23135  snclseqg  24081  ust0  24185  0lt1s  27804  cuteq1  27809  made0  27855  cofcutr  27916  mulsrid  28105  n0cut  28326  n0fincut  28347  zcuts  28399  twocut  28415  halfcut  28450  addhalfcut  28451  pw2cut2  28454  domnprodeq0  33337  grplsm0l  33463  rprmdvdsprod  33594  esum2dlem  34236  eulerpartlemgh  34522  eldm3  35943  poimirlem28  37969  heiborlem3  38134  tfsconcatrn  43770  nregmodel  45444  stgr1  48437  setc1onsubc  50077