Theorem bnj1294 31014

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 2946	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
4		sp 2091	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) → (𝑥 ∈ 𝐴 → 𝜓))
5	4	impcom 445	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) → 𝜓)
6	3, 5	sylan2b 491	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝜓) → 𝜓)
7	1, 2, 6	syl2anc 694	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1521   ∈ wcel 2030  ∀wral 2941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087

This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-ral 2946

This theorem is referenced by:  bnj1379  31027  bnj1121  31179  bnj1279  31212  bnj1286  31213  bnj1296  31215  bnj1421  31236  bnj1450  31244  bnj1489  31250  bnj1501  31261  bnj1523  31265