Theorem rexeq 2674

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 2319	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2319	. 2 ⊢ Ⅎ𝑥𝐵
3	1, 2	rexeqf 2670	1 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ↔ 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:  rexeqi  2678  rexeqdv  2680  rexeqbi1dv  2682  unieq  3819  bnd2  4174  exss  4228  qseq1  6583  finexdc  6902  supeq1  6985  isomni  7134  ismkv  7151  sup3exmid  8914  exmidunben  12427  neifval  13643  cnprcl2k  13709  bj-nn0sucALT  14733  strcoll2  14738  strcollnft  14739  strcollnfALT  14741  sscoll2  14743