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Theorem rexrab 3000
 Description: Existential quantification over a class abstraction. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Mario Carneiro, 3-Sep-2015.)
Hypothesis
Ref Expression
ralab.1 (y = x → (φψ))
Assertion
Ref Expression
rexrab (x {y A φ}χx A (ψ χ))
Distinct variable groups:   x,y   y,A   ψ,y
Allowed substitution hints:   φ(x,y)   ψ(x)   χ(x,y)   A(x)

Proof of Theorem rexrab
StepHypRef Expression
1 ralab.1 . . . . 5 (y = x → (φψ))
21elrab 2994 . . . 4 (x {y A φ} ↔ (x A ψ))
32anbi1i 676 . . 3 ((x {y A φ} χ) ↔ ((x A ψ) χ))
4 anass 630 . . 3 (((x A ψ) χ) ↔ (x A (ψ χ)))
53, 4bitri 240 . 2 ((x {y A φ} χ) ↔ (x A (ψ χ)))
65rexbii2 2643 1 (x {y A φ}χx A (ψ χ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∈ wcel 1710  ∃wrex 2615  {crab 2618 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-rab 2623  df-v 2861 This theorem is referenced by:  phialllem1  4616
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