Theorem dvdemo2 5086

Description: Demonstration of a theorem (scheme) that requires (meta)variables 𝑥 and 𝑧 to be distinct, but no others. It bundles the theorem schemes ∃𝑥(𝑥 = 𝑥 → 𝑧 ∈ 𝑥) and ∃𝑥(𝑥 = 𝑦 → 𝑦 ∈ 𝑥). Compare dvdemo1 5085. (Contributed by NM, 1-Dec-2006.)

Assertion

Ref	Expression
dvdemo2	⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥)

Distinct variable group:   𝑥,𝑧

Proof of Theorem dvdemo2

Step	Hyp	Ref	Expression
1		el 5081	. 2 ⊢ ∃𝑥 𝑧 ∈ 𝑥
2		ax-1 6	. 2 ⊢ (𝑧 ∈ 𝑥 → (𝑥 = 𝑦 → 𝑧 ∈ 𝑥))
3	1, 2	eximii 1880	1 ⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥)

Syntax hints:   → wi 4  ∃wex 1823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-pow 5077

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824

This theorem is referenced by: (None)