Theorem sb8e 2538

Description: Substitution of variable in existential quantifier. Usage of this theorem is discouraged because it depends on ax-13 2380. For a version requiring disjoint variables, but fewer axioms, see sb8ev 2364. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.) (New usage is discouraged.)

Hypothesis

Ref	Expression
sb8.1	⊢ Ⅎ𝑦𝜑

Assertion

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

Proof of Theorem sb8e

Step	Hyp	Ref	Expression
1		sb8.1	. 2 ⊢ Ⅎ𝑦𝜑
2	1	nfs1 2507	. 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
3		sbequ12 2251	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	1, 2, 3	cbvex 2407	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 209  ∃wex 1782  Ⅎwnf 1786  [wsb 2070

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 1912  ax-6 1971  ax-7 2016  ax-10 2143  ax-11 2159  ax-12 2176  ax-13 2380

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1783  df-nf 1787  df-sb 2071

This theorem is referenced by:  2sb8e  2553  sb8mo  2622  bnj985  32468  exlimddvfi  35876  pm11.58  41513