Theorem sbimi 1652

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 13	. . 3 ⊢ ((x = y → φ) → (x = y → ψ))
3	1	anim2i 552	. . . 4 ⊢ ((x = y ∧ φ) → (x = y ∧ ψ))
4	3	eximi 1576	. . 3 ⊢ (∃x(x = y ∧ φ) → ∃x(x = y ∧ ψ))
5	2, 4	anim12i 549	. 2 ⊢ (((x = y → φ) ∧ ∃x(x = y ∧ φ)) → ((x = y → ψ) ∧ ∃x(x = y ∧ ψ)))
6		df-sb 1649	. 2 ⊢ ([y / x]φ ↔ ((x = y → φ) ∧ ∃x(x = y ∧ φ)))
7		df-sb 1649	. 2 ⊢ ([y / x]ψ ↔ ((x = y → ψ) ∧ ∃x(x = y ∧ ψ)))
8	5, 6, 7	3imtr4i 257	1 ⊢ ([y / x]φ → [y / x]ψ)

This proof depends on syntax axioms:   → wi 4   ∧ wa 358  ∃wex 1541  [wsb 1648

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-sb 1649

This theorem is used by:  sbbii  1653  sb6f  2039  hbsb3  2043  sbi2  2064  sbco  2083  sbidm  2085  sbal1  2126  sbal  2127