Theorem bj-dvdemo1 31990

Description: Remove dependency on ax-13 2234 from dvdemo1 4829 (this removal is noteworthy since dvdemo1 4829 and dvdemo2 4830 illustrate the phenomenon of bundling). (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-dvdemo1	⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥)

Distinct variable group:   𝑥,𝑦

Proof of Theorem bj-dvdemo1

Step	Hyp	Ref	Expression
1		bj-dtru 31985	. . 3 ⊢ ¬ ∀𝑥 𝑥 = 𝑦
2		exnal 1744	. . 3 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
3	1, 2	mpbir 220	. 2 ⊢ ∃𝑥 ¬ 𝑥 = 𝑦
4		pm2.21 119	. 2 ⊢ (¬ 𝑥 = 𝑦 → (𝑥 = 𝑦 → 𝑧 ∈ 𝑥))
5	3, 4	eximii 1754	1 ⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-nul 4717  ax-pow 4769

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701

This theorem is referenced by: (None)