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Theorem sb4a 2385
Description: A version of sb4 2384 that doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
Assertion
Ref Expression
sb4a ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))

Proof of Theorem sb4a
StepHypRef Expression
1 sb1 1940 . 2 ([𝑦 / 𝑥]∀𝑦𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑))
2 equs5a 2376 . 2 (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
31, 2syl 17 1 ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1521  wex 1744  [wsb 1937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750  df-sb 1938
This theorem is referenced by:  hbsb2a  2389  sb6f  2413  bj-hbsb2av  32885
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