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Theorem sb2 2425
Description: One direction of a simplified definition of substitution. The converse requires either a disjoint variable condition (sb6 2038) or a non-freeness hypothesis (sb6f 2459). (Contributed by NM, 13-May-1993.) Revise df-sb 2017. (Revised by Wolf Lammen, 26-Jul-2023.)
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 1779 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦𝜑) → ∀𝑥𝜑))
3 stdpc4 2020 . . 3 (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
42, 3syl6 35 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑))
5 sb4b 2424 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
65biimprd 240 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑))
74, 6pm2.61i 177 1 (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1506  [wsb 2016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-10 2080  ax-12 2107  ax-13 2302
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-ex 1744  df-nf 1748  df-sb 2017
This theorem is referenced by:  sb3  2426  hbsb2  2443  hbsb2a  2445  hbsb2e  2447  equsb1  2452  equsb2  2453  dfsb2  2454  sbequiOLD  2456  sb6f  2459  sbi1OLD  2464  sbeqal1  40181
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