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Theorem spimfw 1646
 Description: Specialization, with additional weakening to allow bundling of x and y. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-1017.) (Proof shortened by Wolf Lammen, 7-Aug-2017.)
Hypotheses
Ref Expression
spimfw.1 ψx ¬ ψ)
spimfw.2 (x = y → (φψ))
Assertion
Ref Expression
spimfw x ¬ x = y → (xφψ))

Proof of Theorem spimfw
StepHypRef Expression
1 spimfw.2 . . 3 (x = y → (φψ))
21speimfw 1645 . 2 x ¬ x = y → (xφxψ))
3 df-ex 1542 . . 3 (xψ ↔ ¬ x ¬ ψ)
4 spimfw.1 . . . 4 ψx ¬ ψ)
54con1i 121 . . 3 x ¬ ψψ)
63, 5sylbi 187 . 2 (xψψ)
72, 6syl6 29 1 x ¬ x = y → (xφψ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  ∃wex 1541 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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:  spimw  1668
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