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Theorem exdistrf 1971
 Description: Distribution of existential quantifiers, with a bound-variable hypothesis saying that y is not free in φ, but x can be free in φ (and there is no distinct variable condition on x and y). (Contributed by Mario Carneiro, 20-Mar-2013.)
Hypothesis
Ref Expression
exdistrf.1 x x = y → Ⅎyφ)
Assertion
Ref Expression
exdistrf (xy(φ ψ) → x(φ yψ))

Proof of Theorem exdistrf
StepHypRef Expression
1 biidd 228 . . . . 5 (x x = y → ((φ ψ) ↔ (φ ψ)))
21drex1 1967 . . . 4 (x x = y → (x(φ ψ) ↔ y(φ ψ)))
32drex2 1968 . . 3 (x x = y → (xx(φ ψ) ↔ xy(φ ψ)))
4 nfe1 1732 . . . . 5 xx(φ ψ)
5419.9 1783 . . . 4 (xx(φ ψ) ↔ x(φ ψ))
6 19.8a 1756 . . . . . 6 (ψyψ)
76anim2i 552 . . . . 5 ((φ ψ) → (φ yψ))
87eximi 1576 . . . 4 (x(φ ψ) → x(φ yψ))
95, 8sylbi 187 . . 3 (xx(φ ψ) → x(φ yψ))
103, 9syl6bir 220 . 2 (x x = y → (xy(φ ψ) → x(φ yψ)))
11 nfnae 1956 . . 3 x ¬ x x = y
12 19.40 1609 . . . 4 (y(φ ψ) → (yφ yψ))
13 exdistrf.1 . . . . . 6 x x = y → Ⅎyφ)
141319.9d 1782 . . . . 5 x x = y → (yφφ))
1514anim1d 547 . . . 4 x x = y → ((yφ yψ) → (φ yψ)))
1612, 15syl5 28 . . 3 x x = y → (y(φ ψ) → (φ yψ)))
1711, 16eximd 1770 . 2 x x = y → (xy(φ ψ) → x(φ yψ)))
1810, 17pm2.61i 156 1 (xy(φ ψ) → x(φ yψ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544 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 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545 This theorem is referenced by:  oprabid  5550
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