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Theorem sb56 2278
Description: Two equivalent ways of expressing the proper substitution of 𝑦 for 𝑥 in 𝜑, when 𝑥 and 𝑦 are distinct, namely, alternate definitions sb5 2277 and sb6 2094. Theorem 6.2 of [Quine] p. 40. The proof does not involve df-sb 2071. The implication "to the left" is equs4 2439 and does not require any disjoint variable condition (but the version with a disjoint variable condition, equs4v 2007, requires fewer axioms). Theorem equs45f 2483 replaces the disjoint variable condition with a non-freeness hypothesis and equs5 2484 replaces it with a distinctor as antecedent. (Contributed by NM, 14-Apr-2008.) Revised to use equsexv 2270 in place of equsex 2441 in order to remove dependency on ax-13 2391. (Revised by BJ, 20-Dec-2020.)
Assertion
Ref Expression
sb56 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sb56
StepHypRef Expression
1 sb5 2277 . 2 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
2 sb6 2094 . 2 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
31, 2bitr3i 280 1 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536  wex 1781  [wsb 2070
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 1912  ax-6 1971  ax-7 2016  ax-10 2146  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 2071
This theorem is referenced by:  sb5OLD  2282  dfsb7  2286  dfsb7OLD  2287  sb4vOLDOLD  2514  sb4vOLDALT  2585  sb5ALT2  2587  mopick  2710  alexeqg  3621  dfdif3  4067  pm13.196a  40903
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