Theorem spc2ev 3586

Description: Existential specialization, using implicit substitution. (Contributed by NM, 3-Aug-1995.)

Hypotheses

Ref	Expression
spc2ev.1	⊢ 𝐴 ∈ V
spc2ev.2	⊢ 𝐵 ∈ V
spc2ev.3	⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
spc2ev	⊢ (𝜓 → ∃𝑥∃𝑦𝜑)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜓,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem spc2ev

Step	Hyp	Ref	Expression
1		spc2ev.1	. 2 ⊢ 𝐴 ∈ V
2		spc2ev.2	. 2 ⊢ 𝐵 ∈ V
3		spc2ev.3	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
4	3	spc2egv 3578	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝜓 → ∃𝑥∃𝑦𝜑))
5	1, 2, 4	mp2an 692	1 ⊢ (𝜓 → ∃𝑥∃𝑦𝜑)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  Vcvv 3459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-clel 2809

This theorem is referenced by:  relop  5830  endisj  9070  dcomex  10459  axcnre  11176  hashle2pr  14493  wlk2f  29556  uhgr3cyclex  30109  qqhval2  33959  satfv1  35331  itg2addnclem3  37643  funop1  47260  cycldlenngric  47889