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Theorem sbimi 1663
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 12 . . 3 ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜓))
31anim2i 328 . . . 4 ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜓))
43eximi 1507 . . 3 (∃𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜓))
52, 4anim12i 325 . 2 (((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)) → ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)))
6 df-sb 1662 . 2 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
7 df-sb 1662 . 2 ([𝑦 / 𝑥]𝜓 ↔ ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)))
85, 6, 73imtr4i 194 1 ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wex 1397  [wsb 1661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-sb 1662
This theorem is referenced by:  sbbii  1664  sb6f  1700  hbsb3  1705  sbidm  1747  sbco  1858  sbcocom  1860  elsb3  1868  elsb4  1869  sbalyz  1891  hbsb4t  1905  moimv  1982  oprcl  3600  peano1  4344  peano2  4345
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