Theorem rexeq 2666

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 2312	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2312	. 2 ⊢ Ⅎ𝑥𝐵
3	1, 2	rexeqf 2662	1 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1348  ∃wrex 2449

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454

This theorem is referenced by:  rexeqi  2670  rexeqdv  2672  rexeqbi1dv  2674  unieq  3803  bnd2  4157  exss  4210  qseq1  6557  finexdc  6876  supeq1  6959  isomni  7108  ismkv  7125  sup3exmid  8860  exmidunben  12368  neifval  12893  cnprcl2k  12959  bj-nn0sucALT  13973  strcoll2  13978  strcollnft  13979  strcollnfALT  13981  sscoll2  13983