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Theorem sbimi 2055
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 603 . . . 4 ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜓))
43eximi 1910 . . 3 (∃𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜓))
52, 4anim12i 600 . 2 (((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)) → ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)))
6 df-sb 2050 . 2 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
7 df-sb 2050 . 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 382  wex 1852  [wsb 2049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853  df-sb 2050
This theorem is referenced by:  sbbii  2056  hbsb3  2511  sb6f  2532  sbi2  2540  sbie  2555  2mo  2700  fmptdF  29796  funcnv4mpt  29810  disjdsct  29820  measiuns  30620  ballotlemodife  30899  bj-hbsb3v  33097  bj-sbidmOLD  33166  mptsnunlem  33522
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