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Theorem rexrab2 3004
 Description: Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
Hypothesis
Ref Expression
ralab2.1 (x = y → (ψχ))
Assertion
Ref Expression
rexrab2 (x {y A φ}ψy A (φ χ))
Distinct variable groups:   x,y   x,A   χ,x   φ,x   ψ,y
Allowed substitution hints:   φ(y)   ψ(x)   χ(y)   A(y)

Proof of Theorem rexrab2
StepHypRef Expression
1 df-rab 2623 . . 3 {y A φ} = {y (y A φ)}
21rexeqi 2812 . 2 (x {y A φ}ψx {y (y A φ)}ψ)
3 ralab2.1 . . 3 (x = y → (ψχ))
43rexab2 3003 . 2 (x {y (y A φ)}ψy((y A φ) χ))
5 anass 630 . . . 4 (((y A φ) χ) ↔ (y A (φ χ)))
65exbii 1582 . . 3 (y((y A φ) χ) ↔ y(y A (φ χ)))
7 df-rex 2620 . . 3 (y A (φ χ) ↔ y(y A (φ χ)))
86, 7bitr4i 243 . 2 (y((y A φ) χ) ↔ y A (φ χ))
92, 4, 83bitri 262 1 (x {y A φ}ψy A (φ χ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   ∈ wcel 1710  {cab 2339  ∃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 This theorem is referenced by: (None)
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