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Theorem hba1w 1707
 Description: Weak version of hba1 1786. See comments for ax6w 1717. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
Hypothesis
Ref Expression
hbn1w.1 (x = y → (φψ))
Assertion
Ref Expression
hba1w (xφxxφ)
Distinct variable groups:   φ,y   ψ,x   x,y
Allowed substitution hints:   φ(x)   ψ(y)

Proof of Theorem hba1w
StepHypRef Expression
1 hbn1w.1 . . . . . . 7 (x = y → (φψ))
21cbvalvw 1702 . . . . . 6 (xφyψ)
32a1i 10 . . . . 5 (x = y → (xφyψ))
43notbid 285 . . . 4 (x = y → (¬ xφ ↔ ¬ yψ))
54spw 1694 . . 3 (x ¬ xφ → ¬ xφ)
65con2i 112 . 2 (xφ → ¬ x ¬ xφ)
74hbn1w 1706 . 2 x ¬ xφx ¬ x ¬ xφ)
81hbn1w 1706 . . . 4 xφx ¬ xφ)
98con1i 121 . . 3 x ¬ xφxφ)
109alimi 1559 . 2 (x ¬ x ¬ xφxxφ)
116, 7, 103syl 18 1 (xφxxφ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀wal 1540 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 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542 This theorem is referenced by: (None)
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