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Theorem sb2 2493
 Description: One direction of a simplified definition of substitution. The converse requires either a disjoint variable condition (sb6 2090) or a non-freeness hypothesis (sb6f 2515). Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 13-May-1993.) Revise df-sb 2070. (Revised by Wolf Lammen, 26-Jul-2023.) (New usage is discouraged.)
Assertion
Ref Expression
sb2 (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)

Proof of Theorem sb2
StepHypRef Expression
1 pm2.27 42 . . . 4 (𝑥 = 𝑦 → ((𝑥 = 𝑦𝜑) → 𝜑))
21al2imi 1817 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦𝜑) → ∀𝑥𝜑))
3 stdpc4 2073 . . 3 (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
42, 3syl6 35 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑))
5 sb4b 2488 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
65biimprd 251 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑))
74, 6pm2.61i 185 1 (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1536  [wsb 2069 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-12 2175  ax-13 2379 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070 This theorem is referenced by:  sb3OLD  2494  hbsb2  2500  hbsb2a  2502  hbsb2e  2504  equsb1  2509  equsb2  2510  dfsb2  2511  sb6f  2515  sbi1OLD  2519  sbeqal1  41117
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