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Theorem equsex 1706
 Description: A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
Hypotheses
Ref Expression
equsex.1 (𝜓 → ∀𝑥𝜓)
equsex.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
equsex (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Proof of Theorem equsex
StepHypRef Expression
1 equsex.1 . . 3 (𝜓 → ∀𝑥𝜓)
2 equsex.2 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
32biimpa 294 . . 3 ((𝑥 = 𝑦𝜑) → 𝜓)
41, 3exlimih 1572 . 2 (∃𝑥(𝑥 = 𝑦𝜑) → 𝜓)
5 a9e 1674 . . 3 𝑥 𝑥 = 𝑦
6 idd 21 . . . . 5 (𝜓 → (𝑥 = 𝑦𝑥 = 𝑦))
72biimprcd 159 . . . . 5 (𝜓 → (𝑥 = 𝑦𝜑))
86, 7jcad 305 . . . 4 (𝜓 → (𝑥 = 𝑦 → (𝑥 = 𝑦𝜑)))
91, 8eximdh 1590 . . 3 (𝜓 → (∃𝑥 𝑥 = 𝑦 → ∃𝑥(𝑥 = 𝑦𝜑)))
105, 9mpi 15 . 2 (𝜓 → ∃𝑥(𝑥 = 𝑦𝜑))
114, 10impbii 125 1 (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1329   = wceq 1331  ∃wex 1468 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-i9 1510  ax-ial 1514 This theorem depends on definitions:  df-bi 116 This theorem is referenced by:  cbvexh  1728  sb56  1857  cleljust  1910  sb10f  1970
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