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Theorem 19.12vv 1898
 Description: Special case of 19.12 1847 where its converse holds. (Contributed by NM, 18-Jul-2001.) (Revised by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
19.12vv (xy(φψ) ↔ yx(φψ))
Distinct variable groups:   ψ,x   φ,y
Allowed substitution hints:   φ(x)   ψ(y)

Proof of Theorem 19.12vv
StepHypRef Expression
1 19.21v 1890 . . 3 (y(φψ) ↔ (φyψ))
21exbii 1582 . 2 (xy(φψ) ↔ x(φyψ))
3 nfv 1619 . . . 4 xψ
43nfal 1842 . . 3 xyψ
5419.36 1871 . 2 (x(φyψ) ↔ (xφyψ))
6 19.36v 1896 . . . 4 (x(φψ) ↔ (xφψ))
76albii 1566 . . 3 (yx(φψ) ↔ y(xφψ))
8 nfv 1619 . . . . 5 yφ
98nfal 1842 . . . 4 yxφ
10919.21 1796 . . 3 (y(xφψ) ↔ (xφyψ))
117, 10bitr2i 241 . 2 ((xφyψ) ↔ yx(φψ))
122, 5, 113bitri 262 1 (xy(φψ) ↔ yx(φψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541 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 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545 This theorem is referenced by: (None)
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