Theorem rexeqi 2678

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 2674	. 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 105   = wceq 1353  ∃wrex 2456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461

This theorem is referenced by:  rexrab2  2906  rexprg  3646  rextpg  3648  rexxp  4773  rexrnmpo  5992  0ct  7108  nninfwlpoimlemg  7175  arch  9175  zproddc  11589  infssuzex  11952  gcdsupex  11960  gcdsupcl  11961  dvdsprmpweqnn  12337  txbas  13843