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Theorem ralab2 2678
 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 {yφ}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 2285 . 2 (x {yφ}ψx(x {yφ} → ψ))
2 nfsab1 2008 . . . 4 y x {yφ}
3 nfv 1398 . . . 4 yψ
42, 3nfim 1442 . . 3 y(x {yφ} → ψ)
5 nfv 1398 . . 3 x(φχ)
6 eleq1 2078 . . . . 5 (x = y → (x {yφ} ↔ y {yφ}))
7 abid 2006 . . . . 5 (y {yφ} ↔ φ)
86, 7syl6bb 185 . . . 4 (x = y → (x {yφ} ↔ φ))
9 ralab2.1 . . . 4 (x = y → (ψχ))
108, 9imbi12d 223 . . 3 (x = y → ((x {yφ} → ψ) ↔ (φχ)))
114, 5, 10cbval 1615 . 2 (x(x {yφ} → ψ) ↔ y(φχ))
121, 11bitri 173 1 (x {yφ}ψy(φχ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1224   ∈ wcel 1370  {cab 2004  ∀wral 2280 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-11 1374  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000 This theorem depends on definitions:  df-bi 110  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-ral 2285 This theorem is referenced by:  ralrab2  2679  ssintab  3602
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