Theorem bj-ex 16358

Description: Existential generalization. (Contributed by BJ, 8-Dec-2019.) Proof modification is discouraged because there are shorter proofs, but using less basic results (like exlimiv 1646 and 19.9ht 1689 or 19.23ht 1545). (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-ex	⊢ (∃𝑥𝜑 → 𝜑)

Distinct variable group:   𝜑,𝑥

Proof of Theorem bj-ex

Step	Hyp	Ref	Expression
1		ax-ie2 1542	. . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑥(𝜑 → 𝜑) ↔ (∃𝑥𝜑 → 𝜑)))
2		ax-17 1574	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3	1, 2	mpg 1499	. 2 ⊢ (∀𝑥(𝜑 → 𝜑) ↔ (∃𝑥𝜑 → 𝜑))
4		id 19	. 2 ⊢ (𝜑 → 𝜑)
5	3, 4	mpgbi 1500	1 ⊢ (∃𝑥𝜑 → 𝜑)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1395  ∃wex 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-gen 1497  ax-ie2 1542  ax-17 1574

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  bj-d0clsepcl  16520  bj-inf2vnlem1  16565  bj-nn0sucALT  16573