Theorem rexeqdv 2556

Description: Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007.)

Hypothesis

Ref	Expression
raleq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
rexeqdv	⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜓))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem rexeqdv

Step	Hyp	Ref	Expression
1		raleq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		rexeq 2550	. 2 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜓))
3	1, 2	syl 14	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ↔ wb 103   = wceq 1284  ∃wrex 2349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354

This theorem is referenced by:  rexeqbidv  2562  rexeqbidva  2564  fnunirn  5427  cbvexfo  5446  genipv  6699  exfzdc  9249  infssuzex  10345