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Theorem equsexd 1708
Description: Deduction form of equsex 1707. (Contributed by Jim Kingdon, 29-Dec-2017.)
Hypotheses
Ref Expression
equsexd.1 (𝜑 → ∀𝑥𝜑)
equsexd.2 (𝜑 → (𝜒 → ∀𝑥𝜒))
equsexd.3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
equsexd (𝜑 → (∃𝑥(𝑥 = 𝑦𝜓) ↔ 𝜒))

Proof of Theorem equsexd
StepHypRef Expression
1 equsexd.1 . . 3 (𝜑 → ∀𝑥𝜑)
2 equsexd.2 . . 3 (𝜑 → (𝜒 → ∀𝑥𝜒))
3 equsexd.3 . . . 4 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
4 bi1 117 . . . . 5 ((𝜓𝜒) → (𝜓𝜒))
54imim2i 12 . . . 4 ((𝑥 = 𝑦 → (𝜓𝜒)) → (𝑥 = 𝑦 → (𝜓𝜒)))
6 pm3.31 260 . . . 4 ((𝑥 = 𝑦 → (𝜓𝜒)) → ((𝑥 = 𝑦𝜓) → 𝜒))
73, 5, 63syl 17 . . 3 (𝜑 → ((𝑥 = 𝑦𝜓) → 𝜒))
81, 2, 7exlimd2 1575 . 2 (𝜑 → (∃𝑥(𝑥 = 𝑦𝜓) → 𝜒))
9 a9e 1675 . . . 4 𝑥 𝑥 = 𝑦
101a1i 9 . . . . . . . . 9 (𝜑 → (𝜑 → ∀𝑥𝜑))
1110, 2jca 304 . . . . . . . 8 (𝜑 → ((𝜑 → ∀𝑥𝜑) ∧ (𝜒 → ∀𝑥𝜒)))
12 anim12 342 . . . . . . . 8 (((𝜑 → ∀𝑥𝜑) ∧ (𝜒 → ∀𝑥𝜒)) → ((𝜑𝜒) → (∀𝑥𝜑 ∧ ∀𝑥𝜒)))
1311, 12syl 14 . . . . . . 7 (𝜑 → ((𝜑𝜒) → (∀𝑥𝜑 ∧ ∀𝑥𝜒)))
14 19.26 1458 . . . . . . 7 (∀𝑥(𝜑𝜒) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜒))
1513, 14syl6ibr 161 . . . . . 6 (𝜑 → ((𝜑𝜒) → ∀𝑥(𝜑𝜒)))
1615anabsi5 569 . . . . 5 ((𝜑𝜒) → ∀𝑥(𝜑𝜒))
17 idd 21 . . . . . . . 8 (𝜒 → (𝑥 = 𝑦𝑥 = 𝑦))
1817a1i 9 . . . . . . 7 (𝜑 → (𝜒 → (𝑥 = 𝑦𝑥 = 𝑦)))
1918imp 123 . . . . . 6 ((𝜑𝜒) → (𝑥 = 𝑦𝑥 = 𝑦))
20 bi2 129 . . . . . . . . 9 ((𝜓𝜒) → (𝜒𝜓))
2120imim2i 12 . . . . . . . 8 ((𝑥 = 𝑦 → (𝜓𝜒)) → (𝑥 = 𝑦 → (𝜒𝜓)))
22 pm2.04 82 . . . . . . . 8 ((𝑥 = 𝑦 → (𝜒𝜓)) → (𝜒 → (𝑥 = 𝑦𝜓)))
233, 21, 223syl 17 . . . . . . 7 (𝜑 → (𝜒 → (𝑥 = 𝑦𝜓)))
2423imp 123 . . . . . 6 ((𝜑𝜒) → (𝑥 = 𝑦𝜓))
2519, 24jcad 305 . . . . 5 ((𝜑𝜒) → (𝑥 = 𝑦 → (𝑥 = 𝑦𝜓)))
2616, 25eximdh 1591 . . . 4 ((𝜑𝜒) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥(𝑥 = 𝑦𝜓)))
279, 26mpi 15 . . 3 ((𝜑𝜒) → ∃𝑥(𝑥 = 𝑦𝜓))
2827ex 114 . 2 (𝜑 → (𝜒 → ∃𝑥(𝑥 = 𝑦𝜓)))
298, 28impbid 128 1 (𝜑 → (∃𝑥(𝑥 = 𝑦𝜓) ↔ 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1330   = wceq 1332  wex 1469
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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-i9 1511  ax-ial 1515
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  cbvexdh  1899
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