Theorem rexsn 4648

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 4642	. 2 ⊢ (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))
4	1, 3	ax-mp 5	1 ⊢ (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  Vcvv 3455  {csn 4589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-v 3457  df-sn 4590

This theorem is referenced by:  elsnres  6020  oarec  8543  snec  8772  zornn0g  10484  fpwwe2lem12  10622  elreal  11111  hashge2el2difr  14514  vdwlem6  17041  pzriprnglem10  21640  pmatcollpw3fi1  22945  restsn  23327  snclseqg  24273  ust0  24377  0lt1s  28005  cuteq1  28010  made0  28056  cofcutr  28117  mulsrid  28306  n0cut  28527  n0fincut  28548  zcuts  28600  twocut  28616  halfcut  28651  addhalfcut  28652  pw2cut2  28655  domnprodeq0  33599  grplsm0l  33712  rprmdvdsprod  33824  esum2dlem  34482  eulerpartlemgh  34768  eldm3  36253  poimirlem28  38299  heiborlem3  38464  tfsconcatrn  44069  nregmodel  45726  stgr1  48726  setc1onsubc  50380