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Theorem sb56 2271
Description: Two equivalent ways of expressing the proper substitution of 𝑦 for 𝑥 in 𝜑, when 𝑥 and 𝑦 are distinct. Theorem 6.2 of [Quine] p. 40. The proof does not involve df-sb 2050. The implication "to the left" is equs4 2445 and does not require any dv condition (but the version with a dv condition, equs4v 2088, requires fewer axioms). Theorem equs45f 2496 replaces the dv condition with a non-freeness hypothesis and equs5 2497 replaces it with a distinctor as antecedent. (Contributed by NM, 14-Apr-2008.) Revised to use equsexv 2265 in place of equsex 2447 in order to remove dependency on ax-13 2408. (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 nfa1 2184 . 2 𝑥𝑥(𝑥 = 𝑦𝜑)
2 ax12v2 2205 . . 3 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
3 sp 2207 . . . 4 (∀𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
43com12 32 . . 3 (𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦𝜑) → 𝜑))
52, 4impbid 202 . 2 (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
61, 5equsexv 2265 1 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wal 1629  wex 1852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-10 2174  ax-12 2203
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-ex 1853  df-nf 1858
This theorem is referenced by:  sb6  2272  sb5  2273  sb6OLD  2576  mopick  2684  alexeqg  3482  dfdif3  3871  bj-sb3v  33092  bj-sb4v  33093  bj-sb6  33103  bj-sb5  33104  pm13.196a  39141
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