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Theorem sb4a 2498
 Description: A version of one implication of sb4b 2488 that does not require a distinctor antecedent. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker sb4av 2242 when possible. (Contributed by NM, 2-Feb-2007.) Revise df-sb 2070. (Revised by Wolf Lammen, 28-Jul-2023.) (New usage is discouraged.)
Assertion
Ref Expression
sb4a ([𝑡 / 𝑥]∀𝑡𝜑 → ∀𝑥(𝑥 = 𝑡𝜑))

Proof of Theorem sb4a
StepHypRef Expression
1 sbequ2 2247 . . . 4 (𝑥 = 𝑡 → ([𝑡 / 𝑥]∀𝑡𝜑 → ∀𝑡𝜑))
21sps 2182 . . 3 (∀𝑥 𝑥 = 𝑡 → ([𝑡 / 𝑥]∀𝑡𝜑 → ∀𝑡𝜑))
3 axc11r 2375 . . . 4 (∀𝑥 𝑥 = 𝑡 → (∀𝑡𝜑 → ∀𝑥𝜑))
4 ala1 1815 . . . 4 (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑡𝜑))
53, 4syl6 35 . . 3 (∀𝑥 𝑥 = 𝑡 → (∀𝑡𝜑 → ∀𝑥(𝑥 = 𝑡𝜑)))
62, 5syld 47 . 2 (∀𝑥 𝑥 = 𝑡 → ([𝑡 / 𝑥]∀𝑡𝜑 → ∀𝑥(𝑥 = 𝑡𝜑)))
7 sb4b 2488 . . 3 (¬ ∀𝑥 𝑥 = 𝑡 → ([𝑡 / 𝑥]∀𝑡𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → ∀𝑡𝜑)))
8 sp 2180 . . . . 5 (∀𝑡𝜑𝜑)
98imim2i 16 . . . 4 ((𝑥 = 𝑡 → ∀𝑡𝜑) → (𝑥 = 𝑡𝜑))
109alimi 1813 . . 3 (∀𝑥(𝑥 = 𝑡 → ∀𝑡𝜑) → ∀𝑥(𝑥 = 𝑡𝜑))
117, 10syl6bi 256 . 2 (¬ ∀𝑥 𝑥 = 𝑡 → ([𝑡 / 𝑥]∀𝑡𝜑 → ∀𝑥(𝑥 = 𝑡𝜑)))
126, 11pm2.61i 185 1 ([𝑡 / 𝑥]∀𝑡𝜑 → ∀𝑥(𝑥 = 𝑡𝜑))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1536  [wsb 2069 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 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-12 2175  ax-13 2379 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070 This theorem is referenced by:  hbsb2a  2502  sb6f  2515
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