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Theorem rexab 2999
 Description: Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.)
Hypothesis
Ref Expression
ralab.1 (y = x → (φψ))
Assertion
Ref Expression
rexab (x {y φ}χx(ψ χ))
Distinct variable groups:   x,y   ψ,y
Allowed substitution hints:   φ(x,y)   ψ(x)   χ(x,y)

Proof of Theorem rexab
StepHypRef Expression
1 df-rex 2620 . 2 (x {y φ}χx(x {y φ} χ))
2 vex 2862 . . . . 5 x V
3 ralab.1 . . . . 5 (y = x → (φψ))
42, 3elab 2985 . . . 4 (x {y φ} ↔ ψ)
54anbi1i 676 . . 3 ((x {y φ} χ) ↔ (ψ χ))
65exbii 1582 . 2 (x(x {y φ} χ) ↔ x(ψ χ))
71, 6bitri 240 1 (x {y φ}χx(ψ χ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   ∈ wcel 1710  {cab 2339  ∃wrex 2615 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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-v 2861 This theorem is referenced by:  opeq  4619  frds  5935
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