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Theorem dfsb7OLD 2287
Description: Obsolete version of dfsb7 2286 as of 3-Sep-2023. (Contributed by NM, 28-Jan-2004.) Revise df-sb 2070. (Revised by BJ, 25-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
dfsb7OLD ([𝑡 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
Distinct variable groups:   𝑦,𝑡   𝑥,𝑦   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑡)

Proof of Theorem dfsb7OLD
StepHypRef Expression
1 df-sb 2070 . 2 ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
2 sb56 2278 . 2 (∃𝑦(𝑦 = 𝑡 ∧ ∀𝑥(𝑥 = 𝑦𝜑)) ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
3 sb56 2278 . . . . 5 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
43bicomi 227 . . . 4 (∀𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥(𝑥 = 𝑦𝜑))
54anbi2i 625 . . 3 ((𝑦 = 𝑡 ∧ ∀𝑥(𝑥 = 𝑦𝜑)) ↔ (𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
65exbii 1849 . 2 (∃𝑦(𝑦 = 𝑡 ∧ ∀𝑥(𝑥 = 𝑦𝜑)) ↔ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
71, 2, 63bitr2i 302 1 ([𝑡 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536  wex 1781  [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 2145  ax-12 2178
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: (None)
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