Theorem bj-dvdemo1 33630

Description: Remove dependency on ax-13 2301 from dvdemo1 5121 (this removal is noteworthy since dvdemo1 5121 and dvdemo2 5122 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 33625	. . 3 ⊢ ¬ ∀𝑥 𝑥 = 𝑦
2		exnal 1789	. . 3 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
3	1, 2	mpbir 223	. 2 ⊢ ∃𝑥 ¬ 𝑥 = 𝑦
4		pm2.21 121	. 2 ⊢ (¬ 𝑥 = 𝑦 → (𝑥 = 𝑦 → 𝑧 ∈ 𝑥))
5	3, 4	eximii 1799	1 ⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1505  ∃wex 1742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-12 2106  ax-nul 5061  ax-pow 5113

This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1743

This theorem is referenced by: (None)