Theorem sbimi 1813

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 1649	. . 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 1812	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
7		df-sb 1812	. 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 1541  [wsb 1811

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-ial 1583

This theorem depends on definitions:  df-bi 117  df-sb 1812

This theorem is referenced by:  sbbii  1814  sb6f  1852  hbsb3  1857  sbidm  1900  sbco  2022  sbcocom  2024  sbalyz  2053  hbsb4t  2067  moimv  2147  elsb1  2210  elsb2  2211  oprcl  3906  peano1  4715  peano2  4716