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Theorem spimehOLD 1821
 Description: Obsolete proof of spimeh 1667 as of 10-Dec-2017. (Contributed by NM, 7-Aug-1994.) (New usage is discouraged.)
Hypotheses
Ref Expression
spimehOLD.1 (φxφ)
spimehOLD.2 (x = z → (φψ))
Assertion
Ref Expression
spimehOLD (φxψ)
Distinct variable group:   x,z
Allowed substitution hints:   φ(x,z)   ψ(x,z)

Proof of Theorem spimehOLD
StepHypRef Expression
1 ax9v 1655 . . . 4 ¬ x ¬ x = z
2 id 19 . . . . . . 7 (φφ)
32hbth 1552 . . . . . . . 8 ((φφ) → x(φφ))
4 hba1 1786 . . . . . . . . 9 (x ¬ ψxx ¬ ψ)
54a1i 10 . . . . . . . 8 ((φφ) → (x ¬ ψxx ¬ ψ))
6 spimehOLD.1 . . . . . . . . . 10 (φxφ)
76hbn 1776 . . . . . . . . 9 φx ¬ φ)
87a1i 10 . . . . . . . 8 ((φφ) → (¬ φx ¬ φ))
93, 5, 8hbimd 1815 . . . . . . 7 ((φφ) → ((x ¬ ψ → ¬ φ) → x(x ¬ ψ → ¬ φ)))
102, 9ax-mp 8 . . . . . 6 ((x ¬ ψ → ¬ φ) → x(x ¬ ψ → ¬ φ))
1110hbn 1776 . . . . 5 (¬ (x ¬ ψ → ¬ φ) → x ¬ (x ¬ ψ → ¬ φ))
12 spimehOLD.2 . . . . . . 7 (x = z → (φψ))
13 sp 1747 . . . . . . 7 (x ¬ ψ → ¬ ψ)
1412, 13nsyli 133 . . . . . 6 (x = z → (x ¬ ψ → ¬ φ))
1514con3i 127 . . . . 5 (¬ (x ¬ ψ → ¬ φ) → ¬ x = z)
1611, 15alrimih 1565 . . . 4 (¬ (x ¬ ψ → ¬ φ) → x ¬ x = z)
171, 16mt3 171 . . 3 (x ¬ ψ → ¬ φ)
1817con2i 112 . 2 (φ → ¬ x ¬ ψ)
19 df-ex 1542 . 2 (xψ ↔ ¬ x ¬ ψ)
2018, 19sylibr 203 1 (φxψ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  ∃wex 1541 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-ex 1542  df-nf 1545 This theorem is referenced by: (None)
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