Theorem bnj1465 34857

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 480	. . 3 ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑉) → 𝜓)
3		bnj1465.2	. . . . 5 ⊢ (𝜓 → ∀𝑥𝜓)
4		bnj1465.1	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
5	3, 4	bnj1464 34856	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
6	5	adantl 481	. . 3 ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑉) → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
7	2, 6	mpbird 257	. 2 ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑉) → [𝐴 / 𝑥]𝜑)
8	7	spesbcd 3829	1 ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑉) → ∃𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2111  [wsbc 3736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-v 3438  df-sbc 3737

This theorem is referenced by:  bnj1463  35067