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Theorem sb5 2300
 Description: Equivalence for substitution. Similar to Theorem 6.1 of [Quine] p. 40. The implication "to the right" is sb1 2067 and does not require any disjoint variable condition. Theorem sb5f 2503 replaces the disjoint variable condition with a non-freeness hypothesis. (Contributed by NM, 18-Aug-1993.)
Assertion
Ref Expression
sb5 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sb5
StepHypRef Expression
1 sb6 2299 . 2 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
2 sb56 2297 . 2 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
31, 2bitr4i 270 1 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385  ∀wal 1651  ∃wex 1875  [wsb 2064 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-10 2185  ax-12 2213 This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-ex 1876  df-nf 1880  df-sb 2065 This theorem is referenced by:  2sb5  2301  sbnv  2326  sb7f  2573  sbelx  2577  sbc2or  3642  sbc5  3658
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