Theorem rexsn 4649

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 4643	. 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 3054  Vcvv 3450  {csn 4592

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 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-v 3452  df-sn 4593

This theorem is referenced by:  elsnres  5995  oarec  8529  snec  8754  zornn0g  10465  fpwwe2lem12  10602  elreal  11091  hashge2el2difr  14453  vdwlem6  16964  pzriprnglem10  21407  pmatcollpw3fi1  22682  restsn  23064  snclseqg  24010  ust0  24114  0slt1s  27748  cuteq1  27753  made0  27792  cofcutr  27839  mulsrid  28023  n0scut  28233  n0sfincut  28253  zscut  28302  1p1e2s  28309  twocut  28316  halfcut  28340  addhalfcut  28341  grplsm0l  33381  rprmdvdsprod  33512  esum2dlem  34089  eulerpartlemgh  34376  eldm3  35755  poimirlem28  37649  heiborlem3  37814  tfsconcatrn  43338  nregmodel  45014  stgr1  47964  setc1onsubc  49595