Theorem immo 1043

Description: "At most one" is preserved through implication (notice wff reversal).

Assertion

Ref	Expression
immo	⊢ (∀x(φ → ψ) → (∃*xψ → ∃*xφ))

Proof of Theorem immo

Step	Hyp	Ref	Expression
1		syl2 17	. . . 4 ⊢ ((φ → ψ) → ((ψ → x = y) → (φ → x = y)))
2	1	19.20ii 692	. . 3 ⊢ (∀x(φ → ψ) → (∀x(ψ → x = y) → ∀x(φ → x = y)))
3	2	19.22dv 947	. 2 ⊢ (∀x(φ → ψ) → (∃y∀x(ψ → x = y) → ∃y∀x(φ → x = y)))
4		ax-17 925	. . 3 ⊢ (ψ → ∀yψ)
5	4	mo2 1026	. 2 ⊢ (∃*xψ ↔ ∃y∀x(ψ → x = y))
6		ax-17 925	. . 3 ⊢ (φ → ∀yφ)
7	6	mo2 1026	. 2 ⊢ (∃*xφ ↔ ∃y∀x(φ → x = y))
8	3, 5, 7	3imtr4g 426	1 ⊢ (∀x(φ → ψ) → (∃*xψ → ∃*xφ))

Syntax hints:   → wi 2  ∀wal 672  ∃wex 678   = weq 797  ∃*wmo 1008

This theorem is referenced by:  euimmo 1045  moan 1046  moor 1048  moexex 1058  2moex 1060  2exeu 1066  2eu1 1067  fvex 2838  caoprmo 3084  th3qlem2 3251

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010

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