Theorem bj-exlimmpbi 34253

Description: Lemma for theorems of the vtoclg 3564 family. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-exlimmpbi.nf	⊢ Ⅎ𝑥𝜓
bj-exlimmpbi.maj	⊢ (𝜒 → (𝜑 ↔ 𝜓))
bj-exlimmpbi.min	⊢ 𝜑

Assertion

Ref	Expression
bj-exlimmpbi	⊢ (∃𝑥𝜒 → 𝜓)

Proof of Theorem bj-exlimmpbi

Step	Hyp	Ref	Expression
1		bj-exlimmpbi.nf	. 2 ⊢ Ⅎ𝑥𝜓
2		bj-exlimmpbi.min	. . 3 ⊢ 𝜑
3		bj-exlimmpbi.maj	. . 3 ⊢ (𝜒 → (𝜑 ↔ 𝜓))
4	2, 3	mpbii 235	. 2 ⊢ (𝜒 → 𝜓)
5	1, 4	exlimi 2216	1 ⊢ (∃𝑥𝜒 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 208  ∃wex 1779  Ⅎwnf 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-12 2176

This theorem depends on definitions:  df-bi 209  df-ex 1780  df-nf 1784

This theorem is referenced by: (None)