Theorem sb6rf 2091

Description: Reversed substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)

Hypothesis

Ref	Expression
sb5rf.1	⊢ Ⅎyφ

Assertion

Ref	Expression
sb6rf	⊢ (φ ↔ ∀y(y = x → [y / x]φ))

Proof of Theorem sb6rf

Step	Hyp	Ref	Expression
1		sb5rf.1	. . 3 ⊢ Ⅎyφ
2		sbequ1 1918	. . . . 5 ⊢ (x = y → (φ → [y / x]φ))
3	2	equcoms 1681	. . . 4 ⊢ (y = x → (φ → [y / x]φ))
4	3	com12 27	. . 3 ⊢ (φ → (y = x → [y / x]φ))
5	1, 4	alrimi 1765	. 2 ⊢ (φ → ∀y(y = x → [y / x]φ))
6		sb2 2023	. . 3 ⊢ (∀y(y = x → [y / x]φ) → [x / y][y / x]φ)
7	1	sbid2 2084	. . 3 ⊢ ([x / y][y / x]φ ↔ φ)
8	6, 7	sylib 188	. 2 ⊢ (∀y(y = x → [y / x]φ) → φ)
9	5, 8	impbii 180	1 ⊢ (φ ↔ ∀y(y = x → [y / x]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:  2sb6rf  2118  eu1  2225