Theorem bj-ex 15408

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 1612 and 19.9ht 1655 or 19.23ht 1511). (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 1508	. . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑥(𝜑 → 𝜑) ↔ (∃𝑥𝜑 → 𝜑)))
2		ax-17 1540	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3	1, 2	mpg 1465	. 2 ⊢ (∀𝑥(𝜑 → 𝜑) ↔ (∃𝑥𝜑 → 𝜑))
4		id 19	. 2 ⊢ (𝜑 → 𝜑)
5	3, 4	mpgbi 1466	1 ⊢ (∃𝑥𝜑 → 𝜑)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1362  ∃wex 1506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-gen 1463  ax-ie2 1508  ax-17 1540

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  bj-d0clsepcl  15571  bj-inf2vnlem1  15616  bj-nn0sucALT  15624