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Theorem sbimi 1652
 Description: Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.)
Hypothesis
Ref Expression
sbimi.1 (φψ)
Assertion
Ref Expression
sbimi ([y / x]φ → [y / x]ψ)

Proof of Theorem sbimi
StepHypRef Expression
1 sbimi.1 . . . 4 (φψ)
21imim2i 13 . . 3 ((x = yφ) → (x = yψ))
31anim2i 552 . . . 4 ((x = y φ) → (x = y ψ))
43eximi 1576 . . 3 (x(x = y φ) → x(x = y ψ))
52, 4anim12i 549 . 2 (((x = yφ) x(x = y φ)) → ((x = yψ) x(x = y ψ)))
6 df-sb 1649 . 2 ([y / x]φ ↔ ((x = yφ) x(x = y φ)))
7 df-sb 1649 . 2 ([y / x]ψ ↔ ((x = yψ) x(x = y ψ)))
85, 6, 73imtr4i 257 1 ([y / x]φ → [y / x]ψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541  [wsb 1648 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  df-sb 1649 This theorem is referenced by:  sbbii  1653  sb6f  2039  hbsb3  2043  sbi2  2064  sbco  2083  sbidm  2085  sbal1  2126  sbal  2127
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