Theorem dfsb7a 2045

Description: An alternate definition of proper substitution df-sb 1809. Similar to dfsb7 2042 in that it involves a dummy variable 𝑧, but expressed in terms of ∀ rather than ∃. For a version which only requires Ⅎ𝑧𝜑 rather than 𝑧 and 𝜑 being distinct, see sb7af 2044. (Contributed by Jim Kingdon, 5-Feb-2018.)

Assertion

Ref	Expression
dfsb7a	⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧   𝜑,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem dfsb7a

Step	Hyp	Ref	Expression
1		nfv 1574	. 2 ⊢ Ⅎ𝑧𝜑
2	1	sb7af 2044	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1393  [wsb 1808

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809

This theorem is referenced by: (None)