Theorem sb7af 2046

Description: An alternate definition of proper substitution df-sb 1811. Similar to dfsb7a 2047 but does not require that 𝜑 and 𝑧 be distinct. Similar to sb7f 2045 in that it involves a dummy variable 𝑧, but expressed in terms of ∀ rather than ∃. (Contributed by Jim Kingdon, 5-Feb-2018.)

Hypothesis

Ref	Expression
sb7af.1	⊢ Ⅎ𝑧𝜑

Assertion

Ref	Expression
sb7af	⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

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

Proof of Theorem sb7af

Step	Hyp	Ref	Expression
1		sb6 1935	. . 3 ⊢ ([𝑧 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑧 → 𝜑))
2	1	sbbii 1813	. 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧]∀𝑥(𝑥 = 𝑧 → 𝜑))
3		sb7af.1	. . 3 ⊢ Ⅎ𝑧𝜑
4	3	sbco2 2018	. 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
5		sb6 1935	. 2 ⊢ ([𝑦 / 𝑧]∀𝑥(𝑥 = 𝑧 → 𝜑) ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))
6	2, 4, 5	3bitr3i 210	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1396  Ⅎwnf 1509  [wsb 1810

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811

This theorem is referenced by:  dfsb7a  2047