Theorem sbimi 1810

Description: Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.)

Hypothesis

Ref	Expression
sbimi.1	⊢ (𝜑 → 𝜓)

Assertion

Ref	Expression
sbimi	⊢ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)

Proof of Theorem sbimi

Step	Hyp	Ref	Expression
1		sbimi.1	. . . 4 ⊢ (𝜑 → 𝜓)
2	1	imim2i 12	. . 3 ⊢ ((𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜓))
3	1	anim2i 342	. . . 4 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 ∧ 𝜓))
4	3	eximi 1646	. . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜓))
5	2, 4	anim12i 338	. 2 ⊢ (((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) → ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)))
6		df-sb 1809	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
7		df-sb 1809	. 2 ⊢ ([𝑦 / 𝑥]𝜓 ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)))
8	5, 6, 7	3imtr4i 201	1 ⊢ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)

Syntax hints:   → wi 4   ∧ wa 104  ∃wex 1538  [wsb 1808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-ial 1580

This theorem depends on definitions:  df-bi 117  df-sb 1809

This theorem is referenced by:  sbbii  1811  sb6f  1849  hbsb3  1854  sbidm  1897  sbco  2019  sbcocom  2021  sbalyz  2050  hbsb4t  2064  moimv  2144  elsb1  2207  elsb2  2208  oprcl  3880  peano1  4683  peano2  4684