Theorem spcdv 2771

Description: Rule of specialization, using implicit substitution. Analogous to rspcdv 2792. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
spcimdv.1	⊢ (𝜑 → 𝐴 ∈ 𝐵)
spcdv.2	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
spcdv	⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝜒,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝐵(𝑥)

Proof of Theorem spcdv

Step	Hyp	Ref	Expression
1		spcimdv.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
2		spcdv.2	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
3	2	biimpd 143	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒))
4	1, 3	spcimdv 2770	1 ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1329   = wceq 1331   ∈ wcel 1480

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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688

This theorem is referenced by: (None)