Theorem bj-ex 12969

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 1577 and 19.9ht 1620 or 19.23ht 1473). (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 1470	. . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑥(𝜑 → 𝜑) ↔ (∃𝑥𝜑 → 𝜑)))
2		ax-17 1506	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3	1, 2	mpg 1427	. 2 ⊢ (∀𝑥(𝜑 → 𝜑) ↔ (∃𝑥𝜑 → 𝜑))
4		id 19	. 2 ⊢ (𝜑 → 𝜑)
5	3, 4	mpgbi 1428	1 ⊢ (∃𝑥𝜑 → 𝜑)

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1329  ∃wex 1468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-gen 1425  ax-ie2 1470  ax-17 1506

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  bj-d0clsepcl  13123  bj-inf2vnlem1  13168  bj-nn0sucALT  13176