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Theorem ax11w 1721
 Description: Weak version of ax-11 1746 from which we can prove any ax-11 1746 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. An instance of the first hypothesis will normally require that x and y be distinct (unless x does not occur in φ). (Contributed by NM, 10-Apr-2017.)
Hypotheses
Ref Expression
ax11w.1 (x = y → (φψ))
ax11w.2 (y = z → (φχ))
Assertion
Ref Expression
ax11w (x = y → (yφx(x = yφ)))
Distinct variable groups:   y,z   ψ,x   φ,z   χ,y
Allowed substitution hints:   φ(x,y)   ψ(y,z)   χ(x,z)

Proof of Theorem ax11w
StepHypRef Expression
1 ax11w.2 . . 3 (y = z → (φχ))
21spw 1694 . 2 (yφφ)
3 ax11w.1 . . 3 (x = y → (φψ))
43ax11wlem 1720 . 2 (x = y → (φx(x = yφ)))
52, 4syl5 28 1 (x = y → (yφx(x = yφ)))
 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:  ax11wdemo  1723
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