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Theorem ralab2 3001
 Description: Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
Hypothesis
Ref Expression
ralab2.1 (x = y → (ψχ))
Assertion
Ref Expression
ralab2 (x {y φ}ψy(φχ))
Distinct variable groups:   x,y   χ,x   φ,x   ψ,y
Allowed substitution hints:   φ(y)   ψ(x)   χ(y)

Proof of Theorem ralab2
StepHypRef Expression
1 df-ral 2619 . 2 (x {y φ}ψx(x {y φ} → ψ))
2 nfsab1 2343 . . . 4 y x {y φ}
3 nfv 1619 . . . 4 yψ
42, 3nfim 1813 . . 3 y(x {y φ} → ψ)
5 nfv 1619 . . 3 x(φχ)
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, 9imbi12d 311 . . 3 (x = y → ((x {y φ} → ψ) ↔ (φχ)))
114, 5, 10cbval 1984 . 2 (x(x {y φ} → ψ) ↔ y(φχ))
121, 11bitri 240 1 (x {y φ}ψy(φχ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2614 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-ral 2619 This theorem is referenced by:  ralrab2  3002  ssintab  3943
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