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Theorem rabbi2dva 3463
 Description: Deduction from a wff to a restricted class abstraction. (Contributed by NM, 14-Jan-2014.)
Hypothesis
Ref Expression
rabbi2dva.1 ((φ x A) → (x Bψ))
Assertion
Ref Expression
rabbi2dva (φ → (AB) = {x A ψ})
Distinct variable groups:   φ,x   x,A   x,B
Allowed substitution hint:   ψ(x)

Proof of Theorem rabbi2dva
StepHypRef Expression
1 elin 3219 . . . 4 (x (AB) ↔ (x A x B))
21abbi2i 2464 . . 3 (AB) = {x (x A x B)}
3 df-rab 2623 . . 3 {x A x B} = {x (x A x B)}
42, 3eqtr4i 2376 . 2 (AB) = {x A x B}
5 rabbi2dva.1 . . 3 ((φ x A) → (x Bψ))
65rabbidva 2850 . 2 (φ → {x A x B} = {x A ψ})
74, 6syl5eq 2397 1 (φ → (AB) = {x A ψ})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  {crab 2618   ∩ cin 3208 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213 This theorem is referenced by: (None)
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