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Theorem sb2 2487
Description: One direction of a simplified definition of substitution. The converse requires either a disjoint variable condition (sb6 2085) or a nonfreeness hypothesis (sb6f 2505). Usage of this theorem is discouraged because it depends on ax-13 2380. (Contributed by NM, 13-May-1993.) Revise df-sb 2065. (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 1813 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦𝜑) → ∀𝑥𝜑))
3 stdpc4 2068 . . 3 (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
42, 3syl6 35 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑))
5 sb4b 2483 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
65biimprd 248 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑))
74, 6pm2.61i 182 1 (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1535  [wsb 2064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2141  ax-12 2178  ax-13 2380
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-ex 1778  df-nf 1782  df-sb 2065
This theorem is referenced by:  hbsb2  2490  hbsb2a  2492  hbsb2e  2494  equsb1  2499  equsb2  2500  dfsb2  2501  sb6f  2505  sbeqal1  44367
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