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Theorem sb2 2380
Description: One direction of a simplified definition of substitution. The converse requires either a dv condition (sb6 2457) or a non-freeness hypothesis (sb6f 2413). (Contributed by NM, 13-May-1993.)
Assertion
Ref Expression
sb2 (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)

Proof of Theorem sb2
StepHypRef Expression
1 sp 2091 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
2 equs4 2326 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
3 df-sb 1938 . 2 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
41, 2, 3sylanbrc 699 1 (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1521  wex 1744  [wsb 1937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-sb 1938
This theorem is referenced by:  stdpc4  2381  sb3  2383  sb4b  2386  hbsb2  2387  hbsb2a  2389  hbsb2e  2391  equsb1  2396  equsb2  2397  dfsb2  2401  sbequi  2403  sb6f  2413  sbi1  2420  sb6  2457  iota4  5907  wl-lem-moexsb  33480  sbeqal1  38915
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