Theorem sb6f 1791

Description: Equivalence for substitution when 𝑦 is not free in 𝜑. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 30-Apr-2008.)

Hypothesis

Ref	Expression
equs45f.1	⊢ (𝜑 → ∀𝑦𝜑)

Assertion

Ref	Expression
sb6f	⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))

Proof of Theorem sb6f

Step	Hyp	Ref	Expression
1		equs45f.1	. . . 4 ⊢ (𝜑 → ∀𝑦𝜑)
2	1	sbimi 1752	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]∀𝑦𝜑)
3		sb4a 1789	. . 3 ⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
4	2, 3	syl 14	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
5		sb2 1755	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑)
6	4, 5	impbii 125	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1341  [wsb 1750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-11 1494  ax-4 1498  ax-i9 1518  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-sb 1751

This theorem is referenced by:  sb5f  1792  sbcof2  1798