Theorem spsbim 2076

Description: Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Assertion

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

Proof of Theorem spsbim

Step	Hyp	Ref	Expression
1		stdpc4 2024	. 2 ⊢ (∀x(φ → ψ) → [y / x](φ → ψ))
2		sbi1 2063	. 2 ⊢ ([y / x](φ → ψ) → ([y / x]φ → [y / x]ψ))
3	1, 2	syl 15	1 ⊢ (∀x(φ → ψ) → ([y / x]φ → [y / x]ψ))

Syntax hints:   → wi 4  ∀wal 1540  [wsb 1648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by: (None)