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Theorem sb4a 2471
Description: A version of one implication of sb4b 2466 that does not require a distinctor antecedent. Usage of this theorem is discouraged because it depends on ax-13 2363. Use the weaker sb4av 2228 when possible. (Contributed by NM, 2-Feb-2007.) Revise df-sb 2060. (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 2233 . . . 4 (𝑥 = 𝑡 → ([𝑡 / 𝑥]∀𝑡𝜑 → ∀𝑡𝜑))
21sps 2170 . . 3 (∀𝑥 𝑥 = 𝑡 → ([𝑡 / 𝑥]∀𝑡𝜑 → ∀𝑡𝜑))
3 axc11r 2357 . . . 4 (∀𝑥 𝑥 = 𝑡 → (∀𝑡𝜑 → ∀𝑥𝜑))
4 ala1 1807 . . . 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 2466 . . 3 (¬ ∀𝑥 𝑥 = 𝑡 → ([𝑡 / 𝑥]∀𝑡𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → ∀𝑡𝜑)))
8 sp 2168 . . . . 5 (∀𝑡𝜑𝜑)
98imim2i 16 . . . 4 ((𝑥 = 𝑡 → ∀𝑡𝜑) → (𝑥 = 𝑡𝜑))
109alimi 1805 . . 3 (∀𝑥(𝑥 = 𝑡 → ∀𝑡𝜑) → ∀𝑥(𝑥 = 𝑡𝜑))
117, 10biimtrdi 252 . 2 (¬ ∀𝑥 𝑥 = 𝑡 → ([𝑡 / 𝑥]∀𝑡𝜑 → ∀𝑥(𝑥 = 𝑡𝜑)))
126, 11pm2.61i 182 1 ([𝑡 / 𝑥]∀𝑡𝜑 → ∀𝑥(𝑥 = 𝑡𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1531  [wsb 2059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-12 2163  ax-13 2363
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ex 1774  df-nf 1778  df-sb 2060
This theorem is referenced by:  hbsb2a  2475  sb6f  2488
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