Theorem sb5OLD 2269

Description: Obsolete version of sb5 2268 as of 21-Sep-2024. (Contributed by NM, 18-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
sb5OLD	⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sb5OLD

Step	Hyp	Ref	Expression
1		nfs1v 2153	. . 3 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
2		sbequ12 2244	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
3	1, 2	equsexv 2260	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ [𝑦 / 𝑥]𝜑)
4	3	bicomi 223	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))

Syntax hints:   ↔ wb 205   ∧ wa 396  ∃wex 1782  [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-10 2137  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ex 1783  df-nf 1787  df-sb 2068

This theorem is referenced by: (None)