Theorem rexeq 2730

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 2373	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2373	. 2 ⊢ Ⅎ𝑥𝐵
3	1, 2	rexeqf 2726	1 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1397  ∃wrex 2510

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2212

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1810  df-cleq 2223  df-clel 2226  df-nfc 2362  df-rex 2515

This theorem is referenced by:  rexeqi  2734  rexeqdv  2736  rexeqbi1dv  2742  unieq  3903  bnd2  4265  exss  4321  qseq1  6757  finexdc  7097  supeq1  7190  isomni  7340  ismkv  7357  sup3exmid  9142  exmidunben  13070  neifval  14893  cnprcl2k  14959  bj-nn0sucALT  16633  strcoll2  16638  strcollnft  16639  strcollnfALT  16641  sscoll2  16643