Theorem sb6rf 2468

Description: Reversed substitution. For a version requiring disjoint variables, but fewer axioms, see sb6rfv 2355. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker sb6rfv 2355 if possible. (Contributed by NM, 1-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 21-Sep-2018.) (New usage is discouraged.)

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		sbequ12r 2245	. . 3 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑))
3	1, 2	equsal 2417	. 2 ⊢ (∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑) ↔ 𝜑)
4	3	bicomi 223	1 ⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537  Ⅎwnf 1786  [wsb 2067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-12 2171  ax-13 2372

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-nf 1787  df-sb 2068

This theorem is referenced by: (None)