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Theorem sb6rf 2091
 Description: Reversed substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypothesis
Ref Expression
sb5rf.1 yφ
Assertion
Ref Expression
sb6rf (φy(y = x → [y / x]φ))

Proof of Theorem sb6rf
StepHypRef Expression
1 sb5rf.1 . . 3 yφ
2 sbequ1 1918 . . . . 5 (x = y → (φ → [y / x]φ))
32equcoms 1681 . . . 4 (y = x → (φ → [y / x]φ))
43com12 27 . . 3 (φ → (y = x → [y / x]φ))
51, 4alrimi 1765 . 2 (φy(y = x → [y / x]φ))
6 sb2 2023 . . 3 (y(y = x → [y / x]φ) → [x / y][y / x]φ)
71sbid2 2084 . . 3 ([x / y][y / x]φφ)
86, 7sylib 188 . 2 (y(y = x → [y / x]φ) → φ)
95, 8impbii 180 1 (φy(y = x → [y / x]φ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544  [wsb 1648 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 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649 This theorem is referenced by:  2sb6rf  2118  eu1  2225
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