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Theorem hbe1w 1708
 Description: Weak version of hbe1 1731. See comments for ax6w 1717. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.)
Hypothesis
Ref Expression
hbn1w.1 (x = y → (φψ))
Assertion
Ref Expression
hbe1w (xφxxφ)
Distinct variable groups:   φ,y   ψ,x   x,y
Allowed substitution hints:   φ(x)   ψ(y)

Proof of Theorem hbe1w
StepHypRef Expression
1 df-ex 1542 . 2 (xφ ↔ ¬ x ¬ φ)
2 hbn1w.1 . . . 4 (x = y → (φψ))
32notbid 285 . . 3 (x = y → (¬ φ ↔ ¬ ψ))
43hbn1w 1706 . 2 x ¬ φx ¬ x ¬ φ)
51, 4hbxfrbi 1568 1 (xφxxφ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀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 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542 This theorem is referenced by: (None)
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