Theorem bnj1294 35494

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1294.1	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
bnj1294.2	⊢ (𝜑 → 𝑥 ∈ 𝐴)

Assertion

Ref	Expression
bnj1294	⊢ (𝜑 → 𝜓)

Proof of Theorem bnj1294

Step	Hyp	Ref	Expression
1		bnj1294.2	. 2 ⊢ (𝜑 → 𝑥 ∈ 𝐴)
2		bnj1294.1	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
3		df-ral 3077	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
4		sp 2219	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) → (𝑥 ∈ 𝐴 → 𝜓))
5	4	impcom 413	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) → 𝜓)
6	3, 5	sylan2b 606	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝜓) → 𝜓)
7	1, 2, 6	syl2anc 596	1 ⊢ (𝜑 → 𝜓)

This proof depends on syntax axioms:   → wi 4  ∀wal 1567   ∈ wcel 2144  ∀wral 3076

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-ral 3077

This theorem is used by:  bnj1379  35507  bnj1121  35662  bnj1279  35695  bnj1286  35696  bnj1296  35698  bnj1421  35719  bnj1489  35733  bnj1501  35744  bnj1523  35748