Theorem rexeq 2625

Description: Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.)

Assertion

Ref	Expression
rexeq	⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rexeq

Step	Hyp	Ref	Expression
1		nfcv 2279	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2279	. 2 ⊢ Ⅎ𝑥𝐵
3	1, 2	rexeqf 2621	1 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1331  ∃wrex 2415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420

This theorem is referenced by:  rexeqi  2629  rexeqdv  2631  rexeqbi1dv  2633  unieq  3740  bnd2  4092  exss  4144  qseq1  6470  finexdc  6789  supeq1  6866  isomni  7001  ismkv  7020  sup3exmid  8708  exmidunben  11928  neifval  12298  cnprcl2k  12364  bj-nn0sucALT  13165  strcoll2  13170  sscoll2  13175