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Theorem sbequ2 2243
Description: An equality theorem for substitution. (Contributed by NM, 16-May-1993.) Revise df-sb 2063. (Revised by BJ, 22-Dec-2020.) (Proof shortened by Wolf Lammen, 3-Feb-2024.)
Assertion
Ref Expression
sbequ2 (𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑𝜑))

Proof of Theorem sbequ2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-sb 2063 . . . 4 ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
21biimpi 217 . . 3 ([𝑡 / 𝑥]𝜑 → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
3 equvinva 2030 . . 3 (𝑥 = 𝑡 → ∃𝑦(𝑥 = 𝑦𝑡 = 𝑦))
4 equcomi 2017 . . . . . 6 (𝑡 = 𝑦𝑦 = 𝑡)
5 sp 2174 . . . . . 6 (∀𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
64, 5imim12i 62 . . . . 5 ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) → (𝑡 = 𝑦 → (𝑥 = 𝑦𝜑)))
76impcomd 412 . . . 4 ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) → ((𝑥 = 𝑦𝑡 = 𝑦) → 𝜑))
87aleximi 1825 . . 3 (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) → (∃𝑦(𝑥 = 𝑦𝑡 = 𝑦) → ∃𝑦𝜑))
92, 3, 8syl2im 40 . 2 ([𝑡 / 𝑥]𝜑 → (𝑥 = 𝑡 → ∃𝑦𝜑))
10 ax5e 1906 . 2 (∃𝑦𝜑𝜑)
119, 10syl6com 37 1 (𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1528  wex 1773  [wsb 2062
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 1904  ax-6 1963  ax-7 2008  ax-12 2169
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-sb 2063
This theorem is referenced by:  stdpc7  2245  sbequ12  2246  sb1OLD  2505  sb4a  2507  dfsb1  2508  dfsb2  2530  sbequiOLD  2532  sbi1OLD  2540  bj-ssbid2  33881  2pm13.193  40753  2pm13.193VD  41104
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