Theorem sb8e 2543

Description: Substitution of variable in existential quantifier. For a version requiring disjoint variables, but fewer axioms, see sb8ev 2367. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)

Hypothesis

Ref	Expression
sb5rf.1	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
sb8e	⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)

Proof of Theorem sb8e

Step	Hyp	Ref	Expression
1		sb5rf.1	. 2 ⊢ Ⅎ𝑦𝜑
2	1	nfs1 2482	. 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
3		sbequ12 2278	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	1, 2, 3	cbvex 2411	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 198  ∃wex 1875  Ⅎwnf 1879  [wsb 2064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-ex 1876  df-nf 1880  df-sb 2065

This theorem is referenced by:  sbnf2OLD  2560  2sb8e  2587  sb8mo  2653  mo3OLD  2659  bnj985  31540  bj-mo3OLD  33327  sbcexf  34405  exlimddvfi  34413  pm11.58  39372