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Theorem sb56 2133
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 1866. The implication "to the left" is equs4 2272 and does not require any dv condition (but the version with a dv condition, equs4v 1915, requires fewer axioms). Theorem equs45f 2333 replaces the dv condition with a non-freeness hypothesis and equs5 2334 replaces it with a distinctor as antecedent. (Contributed by NM, 14-Apr-2008.) Revised to use equsexv 2092 in place of equsex 2275 in order to remove dependency on ax-13 2228. (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 2013 . 2 𝑥𝑥(𝑥 = 𝑦𝜑)
2 ax12v2 2033 . . 3 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
3 sp 2038 . . . 4 (∀𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
43com12 32 . . 3 (𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦𝜑) → 𝜑))
52, 4impbid 200 . 2 (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
61, 5equsexv 2092 1 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wal 1472  wex 1694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-12 2031
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-ex 1695  df-nf 1700
This theorem is referenced by:  sb6  2412  sb5  2413  mopick  2518  alexeqg  3297  bj-sb6  31762  bj-sb5  31763  pm13.196a  37436
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