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Theorem spw 1694
 Description: Weak version of specialization scheme sp 1747. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 1747 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 1747 having no wff metavariables and mutually distinct setvar variables (see ax11wdemo 1723 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 1747 are spfw 1691 (minimal distinct variable requirements), spnfw 1670 (when x is not free in ¬ φ), spvw 1666 (when x does not appear in φ), sptruw 1671 (when φ is true), and spfalw 1672 (when φ is false). (Contributed by NM, 9-Apr-2017.)
Hypothesis
Ref Expression
spw.1 (x = y → (φψ))
Assertion
Ref Expression
spw (xφφ)
Distinct variable groups:   x,y   ψ,x   φ,y
Allowed substitution hints:   φ(x)   ψ(y)

Proof of Theorem spw
StepHypRef Expression
1 ax-17 1616 . . 3 (xφyxφ)
2 ax-5 1557 . . 3 (y(xφψ) → (yxφyψ))
3 spw.1 . . . . . 6 (x = y → (φψ))
43biimprd 214 . . . . 5 (x = y → (ψφ))
54equcoms 1681 . . . 4 (y = x → (ψφ))
65spimvw 1669 . . 3 (yψφ)
71, 2, 6syl56 30 . 2 (y(xφψ) → (xφφ))
83biimpd 198 . . 3 (x = y → (φψ))
98spimvw 1669 . 2 (xφψ)
107, 9mpg 1548 1 (xφφ)
 Colors of variables: wff setvar class Syntax hints:   → 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:  spvwOLD  1695  spfalwOLD  1699  hba1w  1707  ax11w  1721
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