Theorem sb5rf 2475

Description: Reversed substitution. Usage of this theorem is discouraged because it depends on ax-13 2380. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 20-Sep-2018.) (New usage is discouraged.)

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	. . 3 ⊢ Ⅎ𝑦𝜑
2		sbequ12r 2253	. . 3 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑))
3	1, 2	equsex 2426	. 2 ⊢ (∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑) ↔ 𝜑)
4	3	bicomi 224	1 ⊢ (𝜑 ↔ ∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑))

Syntax hints:   ↔ wb 206   ∧ wa 395  ∃wex 1777  Ⅎwnf 1781  [wsb 2064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-12 2178  ax-13 2380

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-nf 1782  df-sb 2065

This theorem is referenced by: (None)