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Theorem equsexd 1599
Description: Deduction form of equsex 1598. (Contributed by Jim Kingdon, 29-Dec-2017.)
Hypotheses
Ref Expression
equsexd.1 (φxφ)
equsexd.2 (φ → (χxχ))
equsexd.3 (φ → (x = y → (ψχ)))
Assertion
Ref Expression
equsexd (φ → (x(x = y ψ) ↔ χ))

Proof of Theorem equsexd
StepHypRef Expression
1 equsexd.1 . . 3 (φxφ)
2 equsexd.2 . . 3 (φ → (χxχ))
3 equsexd.3 . . . 4 (φ → (x = y → (ψχ)))
4 bi1 111 . . . . 5 ((ψχ) → (ψχ))
54imim2i 12 . . . 4 ((x = y → (ψχ)) → (x = y → (ψχ)))
6 pm3.31 249 . . . 4 ((x = y → (ψχ)) → ((x = y ψ) → χ))
73, 5, 63syl 17 . . 3 (φ → ((x = y ψ) → χ))
81, 2, 7exlimd2 1468 . 2 (φ → (x(x = y ψ) → χ))
9 a9e 1568 . . . 4 x x = y
101a1i 9 . . . . . . . . 9 (φ → (φxφ))
1110, 2jca 290 . . . . . . . 8 (φ → ((φxφ) (χxχ)))
12 prth 326 . . . . . . . 8 (((φxφ) (χxχ)) → ((φ χ) → (xφ xχ)))
1311, 12syl 14 . . . . . . 7 (φ → ((φ χ) → (xφ xχ)))
14 19.26 1350 . . . . . . 7 (x(φ χ) ↔ (xφ xχ))
1513, 14syl6ibr 151 . . . . . 6 (φ → ((φ χ) → x(φ χ)))
1615anabsi5 500 . . . . 5 ((φ χ) → x(φ χ))
17 idd 21 . . . . . . . 8 (χ → (x = yx = y))
1817a1i 9 . . . . . . 7 (φ → (χ → (x = yx = y)))
1918imp 115 . . . . . 6 ((φ χ) → (x = yx = y))
20 bi2 121 . . . . . . . . 9 ((ψχ) → (χψ))
2120imim2i 12 . . . . . . . 8 ((x = y → (ψχ)) → (x = y → (χψ)))
22 pm2.04 76 . . . . . . . 8 ((x = y → (χψ)) → (χ → (x = yψ)))
233, 21, 223syl 17 . . . . . . 7 (φ → (χ → (x = yψ)))
2423imp 115 . . . . . 6 ((φ χ) → (x = yψ))
2519, 24jcad 291 . . . . 5 ((φ χ) → (x = y → (x = y ψ)))
2616, 25eximdh 1484 . . . 4 ((φ χ) → (x x = yx(x = y ψ)))
279, 26mpi 15 . . 3 ((φ χ) → x(x = y ψ))
2827ex 108 . 2 (φ → (χx(x = y ψ)))
298, 28impbid 120 1 (φ → (x(x = y ψ) ↔ χ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1226   = wceq 1228  wex 1362
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 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381  ax-i9 1404  ax-ial 1409
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  cbvexdh  1783
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