Theorem sbimi 1872

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 16	. . 3 ⊢ ((𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜓))
3	1	anim2i 590	. . . 4 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 ∧ 𝜓))
4	3	eximi 1751	. . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜓))
5	2, 4	anim12i 587	. 2 ⊢ (((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) → ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)))
6		df-sb 1867	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
7		df-sb 1867	. 2 ⊢ ([𝑦 / 𝑥]𝜓 ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)))
8	5, 6, 7	3imtr4i 279	1 ⊢ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)

Syntax hints:   → wi 4   ∧ wa 382  ∃wex 1694  [wsb 1866

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727

This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695  df-sb 1867

This theorem is referenced by:  sbbii  1873  hbsb3  2351  sb6f  2372  sbi2  2380  sbie  2395  2mo  2538  fmptdF  28670  funcnv4mpt  28687  disjdsct  28697  measiuns  29441  ballotlemodife  29720  bj-hbsb3v  31783  bj-sbieOLD  31854  bj-sbidmOLD  31855  mptsnunlem  32185