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Theorem sb2 2510
Description: One direction of a simplified definition of substitution. The converse requires either a dv condition (sb6 2284) or a non-freeness hypothesis (sb6f 2544). (Contributed by NM, 13-May-1993.)
Assertion
Ref Expression
sb2 (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)

Proof of Theorem sb2
StepHypRef Expression
1 sp 2218 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
2 equs4 2457 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
3 df-sb 2061 . 2 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
41, 2, 3sylanbrc 574 1 (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1635  wex 1859  [wsb 2060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-12 2214  ax-13 2420
This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1860  df-sb 2061
This theorem is referenced by:  stdpc4  2511  sb3  2513  sb4b  2517  hbsb2  2518  hbsb2a  2520  hbsb2e  2522  equsb1  2527  equsb2  2528  dfsb2  2532  sbequi  2534  sb6f  2544  sbi1  2551  sb6OLD  2588  iota4  6082  wl-lem-moexsb  33664  sbeqal1  39098  absnsb  41651
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