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Theorem sb5 2100
 Description: Equivalence for substitution. Similar to Theorem 6.1 of [Quine] p. 40. (Contributed by NM, 18-Aug-1993.)
Assertion
Ref Expression
sb5 ([y / x]φx(x = y φ))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem sb5
StepHypRef Expression
1 sb6 2099 . 2 ([y / x]φx(x = yφ))
2 sb56 2098 . 2 (x(x = y φ) ↔ x(x = yφ))
31, 2bitr4i 243 1 ([y / x]φx(x = y φ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541  [wsb 1648 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  df-sb 1649 This theorem is referenced by:  2sb5  2112  dfsb7  2119  sb7f  2120  sbelx  2124  sbc2or  3054  sbc5  3070
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