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Theorem equsex 1632
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 284 . . 3 ((𝑥 = 𝑦𝜑) → 𝜓)
41, 3exlimih 1500 . 2 (∃𝑥(𝑥 = 𝑦𝜑) → 𝜓)
5 a9e 1602 . . 3 𝑥 𝑥 = 𝑦
6 idd 21 . . . . 5 (𝜓 → (𝑥 = 𝑦𝑥 = 𝑦))
72biimprcd 153 . . . . 5 (𝜓 → (𝑥 = 𝑦𝜑))
86, 7jcad 295 . . . 4 (𝜓 → (𝑥 = 𝑦 → (𝑥 = 𝑦𝜑)))
91, 8eximdh 1518 . . 3 (𝜓 → (∃𝑥 𝑥 = 𝑦 → ∃𝑥(𝑥 = 𝑦𝜑)))
105, 9mpi 15 . 2 (𝜓 → ∃𝑥(𝑥 = 𝑦𝜑))
114, 10impbii 121 1 (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257   = wceq 1259  wex 1397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-i9 1439  ax-ial 1443
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  cbvexh  1654  sb56  1781  cleljust  1829  sb10f  1887
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