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Theorem reuss2 3535
 Description: Transfer uniqueness to a smaller subclass. (Contributed by NM, 20-Oct-2005.)
Assertion
Ref Expression
reuss2 (((A B x A (φψ)) (x A φ ∃!x B ψ)) → ∃!x A φ)
Distinct variable groups:   x,A   x,B
Allowed substitution hints:   φ(x)   ψ(x)

Proof of Theorem reuss2
StepHypRef Expression
1 df-rex 2620 . . 3 (x A φx(x A φ))
2 df-reu 2621 . . 3 (∃!x B ψ∃!x(x B ψ))
31, 2anbi12i 678 . 2 ((x A φ ∃!x B ψ) ↔ (x(x A φ) ∃!x(x B ψ)))
4 df-ral 2619 . . . . . . 7 (x A (φψ) ↔ x(x A → (φψ)))
5 ssel 3267 . . . . . . . . . . . . . 14 (A B → (x Ax B))
6 prth 554 . . . . . . . . . . . . . 14 (((x Ax B) (φψ)) → ((x A φ) → (x B ψ)))
75, 6sylan 457 . . . . . . . . . . . . 13 ((A B (φψ)) → ((x A φ) → (x B ψ)))
87exp4b 590 . . . . . . . . . . . 12 (A B → ((φψ) → (x A → (φ → (x B ψ)))))
98com23 72 . . . . . . . . . . 11 (A B → (x A → ((φψ) → (φ → (x B ψ)))))
109a2d 23 . . . . . . . . . 10 (A B → ((x A → (φψ)) → (x A → (φ → (x B ψ)))))
1110imp4a 572 . . . . . . . . 9 (A B → ((x A → (φψ)) → ((x A φ) → (x B ψ))))
1211alimdv 1621 . . . . . . . 8 (A B → (x(x A → (φψ)) → x((x A φ) → (x B ψ))))
1312imp 418 . . . . . . 7 ((A B x(x A → (φψ))) → x((x A φ) → (x B ψ)))
144, 13sylan2b 461 . . . . . 6 ((A B x A (φψ)) → x((x A φ) → (x B ψ)))
15 euimmo 2253 . . . . . 6 (x((x A φ) → (x B ψ)) → (∃!x(x B ψ) → ∃*x(x A φ)))
1614, 15syl 15 . . . . 5 ((A B x A (φψ)) → (∃!x(x B ψ) → ∃*x(x A φ)))
17 eu5 2242 . . . . . 6 (∃!x(x A φ) ↔ (x(x A φ) ∃*x(x A φ)))
1817simplbi2 608 . . . . 5 (x(x A φ) → (∃*x(x A φ) → ∃!x(x A φ)))
1916, 18syl9 66 . . . 4 ((A B x A (φψ)) → (x(x A φ) → (∃!x(x B ψ) → ∃!x(x A φ))))
2019imp32 422 . . 3 (((A B x A (φψ)) (x(x A φ) ∃!x(x B ψ))) → ∃!x(x A φ))
21 df-reu 2621 . . 3 (∃!x A φ∃!x(x A φ))
2220, 21sylibr 203 . 2 (((A B x A (φψ)) (x(x A φ) ∃!x(x B ψ))) → ∃!x A φ)
233, 22sylan2b 461 1 (((A B x A (φψ)) (x A φ ∃!x B ψ)) → ∃!x A φ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541   ∈ wcel 1710  ∃!weu 2204  ∃*wmo 2205  ∀wral 2614  ∃wrex 2615  ∃!wreu 2616   ⊆ wss 3257 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-reu 2621  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259 This theorem is referenced by:  reuss  3536  reuun1  3537
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