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Theorem sb56 2136
 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 1868. The implication "to the left" is equs4 2278 and does not require any dv condition (but the version with a dv condition, equs4v 1917, requires fewer axioms). Theorem equs45f 2338 replaces the dv condition with a non-freeness hypothesis and equs5 2339 replaces it with a distinctor as antecedent. (Contributed by NM, 14-Apr-2008.) Revised to use equsexv 2095 in place of equsex 2281 in order to remove dependency on ax-13 2234. (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 2015 . 2 𝑥𝑥(𝑥 = 𝑦𝜑)
2 ax12v2 2036 . . 3 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
3 sp 2041 . . . 4 (∀𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
43com12 32 . . 3 (𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦𝜑) → 𝜑))
52, 4impbid 201 . 2 (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
61, 5equsexv 2095 1 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ex 1696  df-nf 1701 This theorem is referenced by:  sb6  2417  sb5  2418  mopick  2523  alexeqg  3302  bj-sb3v  31944  bj-sb4v  31945  bj-sb6  31955  bj-sb5  31956  pm13.196a  37637
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