Theorem sb6f 2039

Description: Equivalence for substitution when y is not free in φ. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)

Hypothesis

Ref	Expression
sb6f.1	⊢ Ⅎyφ

Assertion

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

Proof of Theorem sb6f

Step	Hyp	Ref	Expression
1		sb6f.1	. . . . 5 ⊢ Ⅎyφ
2	1	nfri 1762	. . . 4 ⊢ (φ → ∀yφ)
3	2	sbimi 1652	. . 3 ⊢ ([y / x]φ → [y / x]∀yφ)
4		sb4a 1923	. . 3 ⊢ ([y / x]∀yφ → ∀x(x = y → φ))
5	3, 4	syl 15	. 2 ⊢ ([y / x]φ → ∀x(x = y → φ))
6		sb2 2023	. 2 ⊢ (∀x(x = y → φ) → [y / x]φ)
7	5, 6	impbii 180	1 ⊢ ([y / x]φ ↔ ∀x(x = y → φ))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544  [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:  sb5f  2040