Theorem bj-ex 16704

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 1651 and 19.9ht 1694 or 19.23ht 1550). (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 1547	. . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑥(𝜑 → 𝜑) ↔ (∃𝑥𝜑 → 𝜑)))
2		ax-17 1579	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3	1, 2	mpg 1504	. 2 ⊢ (∀𝑥(𝜑 → 𝜑) ↔ (∃𝑥𝜑 → 𝜑))
4		id 19	. 2 ⊢ (𝜑 → 𝜑)
5	3, 4	mpgbi 1505	1 ⊢ (∃𝑥𝜑 → 𝜑)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1400  ∃wex 1545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-gen 1502  ax-ie2 1547  ax-17 1579

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  bj-d0clsepcl  16865  bj-inf2vnlem1  16910  bj-nn0sucALT  16918