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Theorem sb1OLD 2508
 Description: Obsolete version of sb1 2504 as of 21-Feb-2024. (Contributed by NM, 13-May-1993.) Revise df-sb 2071. (Revised by Wolf Lammen, 29-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sb1OLD ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))

Proof of Theorem sb1OLD
StepHypRef Expression
1 sbequ2 2251 . . . 4 (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑𝜑))
2 19.8a 2181 . . . . 5 ((𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
32ex 416 . . . 4 (𝑥 = 𝑦 → (𝜑 → ∃𝑥(𝑥 = 𝑦𝜑)))
41, 3syld 47 . . 3 (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑)))
54sps 2185 . 2 (∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑)))
6 sb4b 2500 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
7 equs4 2439 . . 3 (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
86, 7syl6bi 256 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑)))
95, 8pm2.61i 185 1 ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1536  ∃wex 1781  [wsb 2070 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 1912  ax-6 1971  ax-7 2016  ax-10 2146  ax-12 2178  ax-13 2391 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2071 This theorem is referenced by: (None)
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