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Theorem rexab2 3003
 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
rexab2 (x {y φ}ψy(φ χ))
Distinct variable groups:   x,y   χ,x   φ,x   ψ,y
Allowed substitution hints:   φ(y)   ψ(x)   χ(y)

Proof of Theorem rexab2
StepHypRef Expression
1 df-rex 2620 . 2 (x {y φ}ψx(x {y φ} ψ))
2 nfsab1 2343 . . . 4 y x {y φ}
3 nfv 1619 . . . 4 yψ
42, 3nfan 1824 . . 3 y(x {y φ} ψ)
5 nfv 1619 . . 3 x(φ χ)
6 eleq1 2413 . . . . 5 (x = y → (x {y φ} ↔ y {y φ}))
7 abid 2341 . . . . 5 (y {y φ} ↔ φ)
86, 7syl6bb 252 . . . 4 (x = y → (x {y φ} ↔ φ))
9 ralab2.1 . . . 4 (x = y → (ψχ))
108, 9anbi12d 691 . . 3 (x = y → ((x {y φ} ψ) ↔ (φ χ)))
114, 5, 10cbvex 1985 . 2 (x(x {y φ} ψ) ↔ y(φ χ))
121, 11bitri 240 1 (x {y φ}ψy(φ χ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ 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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-rex 2620 This theorem is referenced by:  rexrab2  3004
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