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Theorem sb1 2493
 Description: One direction of a simplified definition of substitution. The converse requires either a disjoint variable condition (sb5 2274) or a non-freeness hypothesis (sb5f 2517). Usage of this theorem is discouraged because it depends on ax-13 2380. Use the weaker sb1v 2093 when possible. (Contributed by NM, 13-May-1993.) Revise df-sb 2071. (Revised by Wolf Lammen, 21-Feb-2024.) (New usage is discouraged.)
Assertion
Ref Expression
sb1 ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))

Proof of Theorem sb1
StepHypRef Expression
1 spsbe 2088 . . 3 ([𝑦 / 𝑥]𝜑 → ∃𝑥𝜑)
2 pm3.2 473 . . . 4 (𝑥 = 𝑦 → (𝜑 → (𝑥 = 𝑦𝜑)))
32aleximi 1834 . . 3 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 → ∃𝑥(𝑥 = 𝑦𝜑)))
41, 3syl5 34 . 2 (∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑)))
5 sb3b 2491 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑)))
65biimpd 232 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑)))
74, 6pm2.61i 185 1 ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1537  ∃wex 1782  [wsb 2070 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-10 2143  ax-12 2176  ax-13 2380 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1783  df-nf 1787  df-sb 2071 This theorem is referenced by:  sb3bOLD  2498  dfsb1  2500  sb4e  2504
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