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Theorem hbn1fw 1705
 Description: Weak version of ax-6 1729 from which we can prove any ax-6 1729 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.)
Hypotheses
Ref Expression
hbn1fw.1 (xφyxφ)
hbn1fw.2 ψx ¬ ψ)
hbn1fw.3 (yψxyψ)
hbn1fw.4 φy ¬ φ)
hbn1fw.5 yψx ¬ yψ)
hbn1fw.6 (x = y → (φψ))
Assertion
Ref Expression
hbn1fw xφx ¬ xφ)
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)   ψ(x,y)

Proof of Theorem hbn1fw
StepHypRef Expression
1 hbn1fw.1 . . . . 5 (xφyxφ)
2 hbn1fw.2 . . . . 5 ψx ¬ ψ)
3 hbn1fw.3 . . . . 5 (yψxyψ)
4 hbn1fw.4 . . . . 5 φy ¬ φ)
5 hbn1fw.6 . . . . 5 (x = y → (φψ))
61, 2, 3, 4, 5cbvalw 1701 . . . 4 (xφyψ)
76biimpri 197 . . 3 (yψxφ)
87con3i 127 . 2 xφ → ¬ yψ)
9 hbn1fw.5 . 2 yψx ¬ yψ)
106biimpi 186 . . . 4 (xφyψ)
1110con3i 127 . . 3 yψ → ¬ xφ)
1211alimi 1559 . 2 (x ¬ yψx ¬ xφ)
138, 9, 123syl 18 1 xφx ¬ xφ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀wal 1540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542 This theorem is referenced by:  hbn1w  1706
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