Theorem bnj1465 32825

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

Hypotheses

Ref	Expression
bnj1465.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
bnj1465.2	⊢ (𝜓 → ∀𝑥𝜓)
bnj1465.3	⊢ (𝜒 → 𝜓)

Assertion

Ref	Expression
bnj1465	⊢ ((𝜒 ∧ 𝐴 ∈ 𝑉) → ∃𝑥𝜑)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑉

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem bnj1465

Step	Hyp	Ref	Expression
1		bnj1465.3	. . . 4 ⊢ (𝜒 → 𝜓)
2	1	adantr 481	. . 3 ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑉) → 𝜓)
3		bnj1465.2	. . . . 5 ⊢ (𝜓 → ∀𝑥𝜓)
4		bnj1465.1	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
5	3, 4	bnj1464 32824	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
6	5	adantl 482	. . 3 ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑉) → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
7	2, 6	mpbird 256	. 2 ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑉) → [𝐴 / 𝑥]𝜑)
8	7	spesbcd 3816	1 ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑉) → ∃𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2106  [wsbc 3716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-v 3434  df-sbc 3717

This theorem is referenced by:  bnj1463  33035