Theorem rexeqi 3325

Description: Equality inference for restricted existential quantifier. (Contributed by Mario Carneiro, 23-Apr-2015.)

Hypothesis

Ref	Expression
raleq1i.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
rexeqi	⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rexeqi

Step	Hyp	Ref	Expression
1		raleq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		rexeq 3322	. 2 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑))
3	1, 2	ax-mp 5	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑)

Syntax hints:   ↔ wb 205   = wceq 1542  ∃wrex 3071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-cleq 2725  df-rex 3072

This theorem is referenced by:  rexrab2  3697  rexprgf  4698  rextpg  4704  rexopabb  5529  rexxp  5843  elidinxpid  6045  elrid  6046  oarec  8562  brttrcl2  9709  ttrcltr  9711  rnttrcl  9717  wwlktovfo  14909  dvdsprmpweqnn  16818  4sqlem12  16889  pmatcollpw3fi1  22290  cmpfi  22912  txbas  23071  xkobval  23090  ustn0  23725  imasdsf1olem  23879  xpsdsval  23887  plyun0  25711  coeeu  25739  1cubr  26347  made0  27369  addsrid  27450  muls01  27571  mulsrid  27572  precsexlemcbv  27655  dfnbgr3  28626  wlkvtxedg  28932  wwlksn0  29148  eucrctshift  29527  adjbdln  31367  elunirnmbfm  33281  satfbrsuc  34388  fmla1  34409  satffunlem2lem2  34428  filnetlem4  35314  rexrabdioph  41580  fnwe2lem2  41841  fourierdlem70  44940  fourierdlem80  44950  pzriprnglem10  46862