Metamath Proof Explorer
Description: Remove from spcimdv dependency on ax-9 , ax-10 , ax-11 ,
ax-13 , ax-ext , df-cleq (and df-nfc , df-v , df-or ,
df-tru , df-nf ). For an even more economical version, see
bj-spcimdvv . (Contributed by BJ, 30-Nov-2020)
(Proof modification is discouraged.)
|
|
Ref |
Expression |
|
Hypotheses |
bj-spcimdv.1 |
⊢ ( 𝜑 → 𝐴 ∈ 𝐵 ) |
|
|
bj-spcimdv.2 |
⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 → 𝜒 ) ) |
|
Assertion |
bj-spcimdv |
⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 → 𝜒 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
bj-spcimdv.1 |
⊢ ( 𝜑 → 𝐴 ∈ 𝐵 ) |
2 |
|
bj-spcimdv.2 |
⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 → 𝜒 ) ) |
3 |
2
|
ex |
⊢ ( 𝜑 → ( 𝑥 = 𝐴 → ( 𝜓 → 𝜒 ) ) ) |
4 |
3
|
alrimiv |
⊢ ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜓 → 𝜒 ) ) ) |
5 |
|
bj-elisset |
⊢ ( 𝐴 ∈ 𝐵 → ∃ 𝑥 𝑥 = 𝐴 ) |
6 |
|
exim |
⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜓 → 𝜒 ) ) → ( ∃ 𝑥 𝑥 = 𝐴 → ∃ 𝑥 ( 𝜓 → 𝜒 ) ) ) |
7 |
5 6
|
syl5 |
⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜓 → 𝜒 ) ) → ( 𝐴 ∈ 𝐵 → ∃ 𝑥 ( 𝜓 → 𝜒 ) ) ) |
8 |
|
19.36v |
⊢ ( ∃ 𝑥 ( 𝜓 → 𝜒 ) ↔ ( ∀ 𝑥 𝜓 → 𝜒 ) ) |
9 |
7 8
|
syl6ib |
⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜓 → 𝜒 ) ) → ( 𝐴 ∈ 𝐵 → ( ∀ 𝑥 𝜓 → 𝜒 ) ) ) |
10 |
4 1 9
|
sylc |
⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 → 𝜒 ) ) |