Theorem rexeq 2744

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 2386	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2386	. 2 ⊢ Ⅎ𝑥𝐵
3	1, 2	rexeqf 2740	1 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1398  ∃wrex 2523

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528

This theorem is referenced by:  rexeqi  2748  rexeqdv  2750  rexeqbi1dv  2756  unieq  3928  bnd2  4291  exss  4348  qseq1  6830  finexdc  7173  supeq1  7290  isomni  7440  ismkv  7457  sup3exmid  9248  exmidunben  13261  neifval  15117  cnprcl2k  15183  bj-nn0sucALT  16860  strcoll2  16865  strcollnft  16866  strcollnfALT  16868  sscoll2  16870