Theorem rexeqi 3143

Description: Equality inference for restricted existential qualifier. (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 3139	. 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 196   = wceq 1483  ∃wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918

This theorem is referenced by:  rexrab2  3374  rexprg  4235  rextpg  4237  rexxp  5264  oarec  7642  wwlktovfo  13701  dvdsprmpweqnn  15589  4sqlem12  15660  pmatcollpw3fi1  20593  cmpfi  21211  txbas  21370  xkobval  21389  ustn0  22024  imasdsf1olem  22178  xpsdsval  22186  plyun0  23953  coeeu  23981  1cubr  24569  dfnbgr3  26236  wlkvtxedg  26540  wwlksn0  26748  wlknwwlksnsur  26776  wlkwwlksur  26783  eucrctshift  27103  adjbdln  28942  elunirnmbfm  30315  filnetlem4  32376  rexrabdioph  37358  fnwe2lem2  37621  fourierdlem70  40393  fourierdlem80  40403