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Theorem sb4a 2465
Description: A version of one implication of sb4b 2459 that does not require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) Revise df-sb 2045. (Revised by Wolf Lammen, 28-Jul-2023.)
Assertion
Ref Expression
sb4a ([𝑡 / 𝑥]∀𝑡𝜑 → ∀𝑥(𝑥 = 𝑡𝜑))

Proof of Theorem sb4a
StepHypRef Expression
1 sbequ2 2216 . . . 4 (𝑥 = 𝑡 → ([𝑡 / 𝑥]∀𝑡𝜑 → ∀𝑡𝜑))
21sps 2150 . . 3 (∀𝑥 𝑥 = 𝑡 → ([𝑡 / 𝑥]∀𝑡𝜑 → ∀𝑡𝜑))
3 axc11r 2345 . . . 4 (∀𝑥 𝑥 = 𝑡 → (∀𝑡𝜑 → ∀𝑥𝜑))
4 ala1 1799 . . . 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 2459 . . 3 (¬ ∀𝑥 𝑥 = 𝑡 → ([𝑡 / 𝑥]∀𝑡𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → ∀𝑡𝜑)))
8 sp 2148 . . . . 5 (∀𝑡𝜑𝜑)
98imim2i 16 . . . 4 ((𝑥 = 𝑡 → ∀𝑡𝜑) → (𝑥 = 𝑡𝜑))
109alimi 1797 . . 3 (∀𝑥(𝑥 = 𝑡 → ∀𝑡𝜑) → ∀𝑥(𝑥 = 𝑡𝜑))
117, 10syl6bi 254 . 2 (¬ ∀𝑥 𝑥 = 𝑡 → ([𝑡 / 𝑥]∀𝑡𝜑 → ∀𝑥(𝑥 = 𝑡𝜑)))
126, 11pm2.61i 183 1 ([𝑡 / 𝑥]∀𝑡𝜑 → ∀𝑥(𝑥 = 𝑡𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1523  [wsb 2044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-10 2114  ax-12 2143  ax-13 2346
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-ex 1766  df-nf 1770  df-sb 2045
This theorem is referenced by:  hbsb2a  2479  sb6f  2493  bj-hbsb2av  33692
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