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Theorem alcomiw 1704
 Description: Weak version of alcom 1737. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.)
Hypothesis
Ref Expression
alcomiw.1 (y = z → (φψ))
Assertion
Ref Expression
alcomiw (xyφyxφ)
Distinct variable groups:   y,z   x,y   φ,z   ψ,y
Allowed substitution hints:   φ(x,y)   ψ(x,z)

Proof of Theorem alcomiw
StepHypRef Expression
1 alcomiw.1 . . . . 5 (y = z → (φψ))
21biimpd 198 . . . 4 (y = z → (φψ))
32cbvalivw 1674 . . 3 (yφzψ)
43alimi 1559 . 2 (xyφxzψ)
5 ax-17 1616 . 2 (xzψyxzψ)
61biimprd 214 . . . . . 6 (y = z → (ψφ))
76equcoms 1681 . . . . 5 (z = y → (ψφ))
87spimvw 1669 . . . 4 (zψφ)
98alimi 1559 . . 3 (xzψxφ)
109alimi 1559 . 2 (yxzψyxφ)
114, 5, 103syl 18 1 (xyφyxφ)
 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:  hbalw  1709  ax7w  1718
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