Theorem bj-dvdemo2 32803

Description: Remove dependency on ax-13 2246 from dvdemo2 4903 (this removal is noteworthy since dvdemo1 4902 and dvdemo2 4903 illustrate the phenomenon of bundling). (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)

Assertion

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

Distinct variable group:   𝑥,𝑧

Proof of Theorem bj-dvdemo2

Step	Hyp	Ref	Expression
1		bj-el 32796	. 2 ⊢ ∃𝑥 𝑧 ∈ 𝑥
2		ax-1 6	. 2 ⊢ (𝑧 ∈ 𝑥 → (𝑥 = 𝑦 → 𝑧 ∈ 𝑥))
3	1, 2	eximii 1764	1 ⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥)

Syntax hints:   → wi 4  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-pow 4843

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)