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Theorem spfw 1691
 Description: Weak version of sp 1747. Uses only Tarski's FOL axiom schemes. Lemma 9 of [KalishMontague] p. 87. This may be the best we can do with minimal distinct variable conditions. TO DO: Do we need this theorem? If not, maybe it should be deleted. (Contributed by NM, 19-Apr-2017.)
Hypotheses
Ref Expression
spfw.1 ψx ¬ ψ)
spfw.2 (xφyxφ)
spfw.3 φy ¬ φ)
spfw.4 (x = y → (φψ))
Assertion
Ref Expression
spfw (xφφ)
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)   ψ(x,y)

Proof of Theorem spfw
StepHypRef Expression
1 spfw.2 . . 3 (xφyxφ)
2 ax-5 1557 . . 3 (y(xφψ) → (yxφyψ))
3 spfw.3 . . . 4 φy ¬ φ)
4 spfw.4 . . . . . 6 (x = y → (φψ))
54biimprd 214 . . . . 5 (x = y → (ψφ))
65equcoms 1681 . . . 4 (y = x → (ψφ))
73, 6spimw 1668 . . 3 (yψφ)
81, 2, 7syl56 30 . 2 (y(xφψ) → (xφφ))
9 spfw.1 . . 3 ψx ¬ ψ)
104biimpd 198 . . 3 (x = y → (φψ))
119, 10spimw 1668 . 2 (xφψ)
128, 11mpg 1548 1 (xφφ)
 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:  spnfwOLD  1692
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