Theorem sb5rf 2090

Description: Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.)

Hypothesis

Ref	Expression
sb5rf.1	⊢ Ⅎyφ

Assertion

Ref	Expression
sb5rf	⊢ (φ ↔ ∃y(y = x ∧ [y / x]φ))

Proof of Theorem sb5rf

Step	Hyp	Ref	Expression
1		sb5rf.1	. . . 4 ⊢ Ⅎyφ
2	1	sbid2 2084	. . 3 ⊢ ([x / y][y / x]φ ↔ φ)
3		sb1 1651	. . 3 ⊢ ([x / y][y / x]φ → ∃y(y = x ∧ [y / x]φ))
4	2, 3	sylbir 204	. 2 ⊢ (φ → ∃y(y = x ∧ [y / x]φ))
5		stdpc7 1917	. . . 4 ⊢ (y = x → ([y / x]φ → φ))
6	5	imp 418	. . 3 ⊢ ((y = x ∧ [y / x]φ) → φ)
7	1, 6	exlimi 1803	. 2 ⊢ (∃y(y = x ∧ [y / x]φ) → φ)
8	4, 7	impbii 180	1 ⊢ (φ ↔ ∃y(y = x ∧ [y / x]φ))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541  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:  2sb5rf  2117