Theorem bj-exlimmp 13485

Description: Lemma for bj-vtoclgf 13492. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-exlimmp.nf	⊢ Ⅎ𝑥𝜓
bj-exlimmp.min	⊢ (𝜒 → 𝜑)

Assertion

Ref	Expression
bj-exlimmp	⊢ (∀𝑥(𝜒 → (𝜑 → 𝜓)) → (∃𝑥𝜒 → 𝜓))

Proof of Theorem bj-exlimmp

Step	Hyp	Ref	Expression
1		nfa1 1528	. 2 ⊢ Ⅎ𝑥∀𝑥(𝜒 → (𝜑 → 𝜓))
2		bj-exlimmp.nf	. 2 ⊢ Ⅎ𝑥𝜓
3		bj-exlimmp.min	. . . . 5 ⊢ (𝜒 → 𝜑)
4		idd 21	. . . . 5 ⊢ (𝜒 → (𝜓 → 𝜓))
5	3, 4	embantd 56	. . . 4 ⊢ (𝜒 → ((𝜑 → 𝜓) → 𝜓))
6	5	a2i 11	. . 3 ⊢ ((𝜒 → (𝜑 → 𝜓)) → (𝜒 → 𝜓))
7	6	sps 1524	. 2 ⊢ (∀𝑥(𝜒 → (𝜑 → 𝜓)) → (𝜒 → 𝜓))
8	1, 2, 7	exlimd 1584	1 ⊢ (∀𝑥(𝜒 → (𝜑 → 𝜓)) → (∃𝑥𝜒 → 𝜓))

Syntax hints:   → wi 4  ∀wal 1340  Ⅎwnf 1447  ∃wex 1479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1434  ax-gen 1436  ax-ie2 1481  ax-4 1497  ax-ial 1521

This theorem depends on definitions:  df-bi 116  df-nf 1448

This theorem is referenced by:  bj-vtoclgft  13491