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Theorem sbimi 2065
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 605 . . . 4 ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜓))
43eximi 1919 . . 3 (∃𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜓))
52, 4anim12i 602 . 2 (((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)) → ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)))
6 df-sb 2060 . 2 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
7 df-sb 2060 . 2 ([𝑦 / 𝑥]𝜓 ↔ ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)))
85, 6, 73imtr4i 283 1 ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wex 1859  [wsb 2059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894
This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1860  df-sb 2060
This theorem is referenced by:  sbbii  2066  hbsb3  2522  sb6f  2543  sbi2  2551  sbie  2566  2mo  2712  fmptdF  29777  funcnv4mpt  29791  disjdsct  29801  measiuns  30599  ballotlemodife  30878  bj-hbsb3v  33069  bj-sbidmOLD  33138  mptsnunlem  33496
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