Theorem sbimi 1644

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 324	. . . 4 ⊢ ((x = y ∧ φ) → (x = y ∧ ψ))
4	3	eximi 1488	. . 3 ⊢ (∃x(x = y ∧ φ) → ∃x(x = y ∧ ψ))
5	2, 4	anim12i 321	. 2 ⊢ (((x = y → φ) ∧ ∃x(x = y ∧ φ)) → ((x = y → ψ) ∧ ∃x(x = y ∧ ψ)))
6		df-sb 1643	. 2 ⊢ ([y / x]φ ↔ ((x = y → φ) ∧ ∃x(x = y ∧ φ)))
7		df-sb 1643	. 2 ⊢ ([y / x]ψ ↔ ((x = y → ψ) ∧ ∃x(x = y ∧ ψ)))
8	5, 6, 7	3imtr4i 190	1 ⊢ ([y / x]φ → [y / x]ψ)

Syntax hints:   → wi 4   ∧ wa 97  ∃wex 1378  [wsb 1642

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-sb 1643

This theorem is referenced by:  sbbii  1645  sb6f  1681  hbsb3  1686  sbidm  1728  sbco  1839  sbcocom  1841  elsb3  1849  elsb4  1850  sbalyz  1872  hbsb4t  1886  moimv  1963  oprcl  3564  peano1  4260  peano2  4261