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Theorem equsex 1613
Description: A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
Hypotheses
Ref Expression
equsex.1 (ψxψ)
equsex.2 (x = y → (φψ))
Assertion
Ref Expression
equsex (x(x = y φ) ↔ ψ)

Proof of Theorem equsex
StepHypRef Expression
1 equsex.1 . . 3 (ψxψ)
2 equsex.2 . . . 4 (x = y → (φψ))
32biimpa 280 . . 3 ((x = y φ) → ψ)
41, 3exlimih 1481 . 2 (x(x = y φ) → ψ)
5 a9e 1583 . . 3 x x = y
6 idd 21 . . . . 5 (ψ → (x = yx = y))
72biimprcd 149 . . . . 5 (ψ → (x = yφ))
86, 7jcad 291 . . . 4 (ψ → (x = y → (x = y φ)))
91, 8eximdh 1499 . . 3 (ψ → (x x = yx(x = y φ)))
105, 9mpi 15 . 2 (ψx(x = y φ))
114, 10impbii 117 1 (x(x = y φ) ↔ ψ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   = wceq 1242  wex 1378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-i9 1420  ax-ial 1424
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  cbvexh  1635  sb56  1762  cleljust  1810  sb10f  1868
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