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Theorem hbalw 1709
 Description: Weak version of hbal 1736. Uses only Tarski's FOL axiom schemes. Unlike hbal 1736, this theorem requires that x and y be distinct i.e. are not bundled. (Contributed by NM, 19-Apr-2017.)
Hypotheses
Ref Expression
hbalw.1 (x = z → (φψ))
hbalw.2 (φxφ)
Assertion
Ref Expression
hbalw (yφxyφ)
Distinct variable groups:   x,z   x,y   φ,z   ψ,x
Allowed substitution hints:   φ(x,y)   ψ(y,z)

Proof of Theorem hbalw
StepHypRef Expression
1 hbalw.2 . . 3 (φxφ)
21alimi 1559 . 2 (yφyxφ)
3 hbalw.1 . . 3 (x = z → (φψ))
43alcomiw 1704 . 2 (yxφxyφ)
52, 4syl 15 1 (yφxyφ)
 Colors of variables: wff setvar class Syntax hints:   → 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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