Theorem sbalv 2129

Description: Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.)

Hypothesis

Ref	Expression
sbalv.1	⊢ ([y / x]φ ↔ ψ)

Assertion

Ref	Expression
sbalv	⊢ ([y / x]∀zφ ↔ ∀zψ)

Distinct variable groups:   x,z   y,z

Allowed substitution hints:   φ(x,y,z)   ψ(x,y,z)

Proof of Theorem sbalv

Step	Hyp	Ref	Expression
1		sbal 2127	. 2 ⊢ ([y / x]∀zφ ↔ ∀z[y / x]φ)
2		sbalv.1	. . 3 ⊢ ([y / x]φ ↔ ψ)
3	2	albii 1566	. 2 ⊢ (∀z[y / x]φ ↔ ∀zψ)
4	1, 3	bitri 240	1 ⊢ ([y / x]∀zφ ↔ ∀zψ)

Syntax hints:   ↔ wb 176  ∀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:  sbmo  2234  sbabel  2515