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Theorem speimfw 1645
Description: Specialization, with additional weakening to allow bundling of x and y. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 5-Aug-2017.)
Hypothesis
Ref Expression
speimfw.2 (x = y → (φψ))
Assertion
Ref Expression
speimfw x ¬ x = y → (xφxψ))

Proof of Theorem speimfw
StepHypRef Expression
1 speimfw.2 . . 3 (x = y → (φψ))
21eximi 1576 . 2 (x x = yx(φψ))
3 df-ex 1542 . 2 (x x = y ↔ ¬ x ¬ x = y)
4 19.35 1600 . 2 (x(φψ) ↔ (xφxψ))
52, 3, 43imtr3i 256 1 x ¬ x = y → (xφxψ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1540  wex 1541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542
This theorem is referenced by:  spimfw  1646  19.2OLD  1700
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