Theorem sbalv 2065

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 2060	. 2 ⊢ ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧[𝑦 / 𝑥]𝜑)
2		sbalv.1	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
3	2	albii 1523	. 2 ⊢ (∀𝑧[𝑦 / 𝑥]𝜑 ↔ ∀𝑧𝜓)
4	1, 3	bitri 184	1 ⊢ ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧𝜓)

Syntax hints:   ↔ wb 105  ∀wal 1400  [wsb 1815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816

This theorem is referenced by:  sbmo  2146  sbabel  2419  peano2  4737