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Theorem sb2 2336
Description: One direction of a simplified definition of substitution. The converse requires either a dv condition (sb6 2413) or a non-freeness hypothesis (sb6f 2369). (Contributed by NM, 13-May-1993.)
Assertion
Ref Expression
sb2 (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)

Proof of Theorem sb2
StepHypRef Expression
1 sp 2039 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
2 equs4 2273 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
3 df-sb 1867 . 2 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
41, 2, 3sylanbrc 694 1 (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wal 1472  wex 1694  [wsb 1866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-12 2032  ax-13 2229
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695  df-sb 1867
This theorem is referenced by:  stdpc4  2337  sb3  2339  sb4b  2342  hbsb2  2343  hbsb2a  2345  hbsb2e  2347  equsb1  2352  equsb2  2353  dfsb2  2357  sbequi  2359  sb6f  2369  sbi1  2376  sb6  2413  iota4  5769  wl-lem-moexsb  32329  sbeqal1  37420
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