Theorem rexsn 4644

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 4638	. 2 ⊢ (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))
4	1, 3	ax-mp 5	1 ⊢ (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2147  ∃wrex 3090  Vcvv 3454  {csn 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-v 3456  df-sn 4589

This theorem is used by:  elsnres  6028  oarec  8608  snec  8837  zornn0g  10622  fpwwe2lem12  10766  elreal  11255  hashge2el2difr  14667  vdwlem6  17208  pzriprnglem10  21851  pmatcollpw3fi1  23161  restsn  23543  snclseqg  24490  ust0  24594  0lt1s  28255  cuteq1  28260  made0  28306  cofcutr  28367  mulsrid  28556  n0cut  28777  n0fincut  28798  zcuts  28850  twocut  28866  halfcut  28901  addhalfcut  28902  pw2cut2  28905  domnprodeq0  33939  grplsm0l  34058  rprmdvdsprod  34170  esum2dlem  34835  eulerpartlemgh  35121  eldm3  36741  poimirlem28  38834  heiborlem3  39013  tfsconcatrn  44605  nregmodel  46271  stgr1  49332  setc1onsubc  51071