Theorem 19.8aOLD 2040

Description: Obsolete proof of 19.8a 2039. Obsolete as of 21-Dec-2020. Can be deleted as soon as the question of why "MM-PA> min exlimiiv" does not give 19.8a 2039 is answered. (Contributed by NM, 9-Jan-1993.) Allow a shortening of sp 2041. (Revised by Wolf Lammen, 13-Jan-2018.) (Proof shortened by Wolf Lammen, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
19.8aOLD	⊢ (𝜑 → ∃𝑥𝜑)

Proof of Theorem 19.8aOLD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax6ev 1877	. 2 ⊢ ∃𝑦 𝑦 = 𝑥
2		ax12v 2035	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
3		ax6ev 1877	. . . . 5 ⊢ ∃𝑥 𝑥 = 𝑦
4		exim 1751	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑))
5	2, 3, 4	syl6mpi 65	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → ∃𝑥𝜑))
6	5	equcoms 1934	. . 3 ⊢ (𝑦 = 𝑥 → (𝜑 → ∃𝑥𝜑))
7	6	exlimiv 1845	. 2 ⊢ (∃𝑦 𝑦 = 𝑥 → (𝜑 → ∃𝑥𝜑))
8	1, 7	ax-mp 5	1 ⊢ (𝜑 → ∃𝑥𝜑)

Syntax hints:   → 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-12 2034

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696

This theorem is referenced by: (None)