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Theorem nfimdOLD 1809
 Description: Obsolete proof of nfimd 1808 as of 29-Dec-2017. (Contributed by Mario Carneiro, 24-Sep-2016.) (New usage is discouraged.)
Hypotheses
Ref Expression
nfimd.1 (φ → Ⅎxψ)
nfimd.2 (φ → Ⅎxχ)
Assertion
Ref Expression
nfimdOLD (φ → Ⅎx(ψχ))

Proof of Theorem nfimdOLD
StepHypRef Expression
1 nfimd.1 . 2 (φ → Ⅎxψ)
2 nfimd.2 . 2 (φ → Ⅎxχ)
3 nfa1 1788 . . . . 5 xx(ψxψ)
4 hbnt 1775 . . . . . 6 (x(ψxψ) → (¬ ψx ¬ ψ))
5 pm2.21 100 . . . . . . . . . . 11 ψ → (ψχ))
65alimi 1559 . . . . . . . . . 10 (x ¬ ψx(ψχ))
76imim2i 13 . . . . . . . . 9 ((¬ ψx ¬ ψ) → (¬ ψx(ψχ)))
87adantr 451 . . . . . . . 8 (((¬ ψx ¬ ψ) (χxχ)) → (¬ ψx(ψχ)))
9 ax-1 5 . . . . . . . . . . 11 (χ → (ψχ))
109alimi 1559 . . . . . . . . . 10 (xχx(ψχ))
1110imim2i 13 . . . . . . . . 9 ((χxχ) → (χx(ψχ)))
1211adantl 452 . . . . . . . 8 (((¬ ψx ¬ ψ) (χxχ)) → (χx(ψχ)))
138, 12jad 154 . . . . . . 7 (((¬ ψx ¬ ψ) (χxχ)) → ((ψχ) → x(ψχ)))
1413ex 423 . . . . . 6 ((¬ ψx ¬ ψ) → ((χxχ) → ((ψχ) → x(ψχ))))
154, 14syl 15 . . . . 5 (x(ψxψ) → ((χxχ) → ((ψχ) → x(ψχ))))
163, 15alimd 1764 . . . 4 (x(ψxψ) → (x(χxχ) → x((ψχ) → x(ψχ))))
1716imp 418 . . 3 ((x(ψxψ) x(χxχ)) → x((ψχ) → x(ψχ)))
18 df-nf 1545 . . . 4 (Ⅎxψx(ψxψ))
19 df-nf 1545 . . . 4 (Ⅎxχx(χxχ))
2018, 19anbi12i 678 . . 3 ((Ⅎxψ xχ) ↔ (x(ψxψ) x(χxχ)))
21 df-nf 1545 . . 3 (Ⅎx(ψχ) ↔ x((ψχ) → x(ψχ)))
2217, 20, 213imtr4i 257 . 2 ((Ⅎxψ xχ) → Ⅎx(ψχ))
231, 2, 22syl2anc 642 1 (φ → Ⅎx(ψχ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544 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-11 1746 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545 This theorem is referenced by: (None)
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