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Theorem sbimi 1883
 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 16 . . 3 ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜓))
31anim2i 592 . . . 4 ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜓))
43eximi 1759 . . 3 (∃𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜓))
52, 4anim12i 589 . 2 (((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)) → ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)))
6 df-sb 1878 . 2 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
7 df-sb 1878 . 2 ([𝑦 / 𝑥]𝜓 ↔ ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)))
85, 6, 73imtr4i 281 1 ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384  ∃wex 1701  [wsb 1877 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734 This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-sb 1878 This theorem is referenced by:  sbbii  1884  hbsb3  2363  sb6f  2384  sbi2  2392  sbie  2407  2mo  2550  fmptdF  29339  funcnv4mpt  29354  disjdsct  29364  measiuns  30103  ballotlemodife  30382  bj-hbsb3v  32457  bj-sbieOLD  32528  bj-sbidmOLD  32529  mptsnunlem  32856
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