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Theorem sbequ2 2246
Description: An equality theorem for substitution. (Contributed by NM, 16-May-1993.) Revise df-sb 2071. (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 2071 . . . 4 ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
21biimpi 219 . . 3 ([𝑡 / 𝑥]𝜑 → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
3 equvinva 2038 . . 3 (𝑥 = 𝑡 → ∃𝑦(𝑥 = 𝑦𝑡 = 𝑦))
4 equcomi 2025 . . . . . 6 (𝑡 = 𝑦𝑦 = 𝑡)
5 sp 2180 . . . . . 6 (∀𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
64, 5imim12i 62 . . . . 5 ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) → (𝑡 = 𝑦 → (𝑥 = 𝑦𝜑)))
76impcomd 415 . . . 4 ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) → ((𝑥 = 𝑦𝑡 = 𝑦) → 𝜑))
87aleximi 1839 . . 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 1920 . 2 (∃𝑦𝜑𝜑)
119, 10syl6com 37 1 (𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wal 1541  wex 1787  [wsb 2070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-12 2175
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-sb 2071
This theorem is referenced by:  stdpc7  2248  sbequ12  2249  sb1OLD  2481  sb4a  2483  dfsb1  2484  dfsb2  2496  bj-ssbid2  34580  2pm13.193  41845  2pm13.193VD  42196
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