Theorem spc2ev 3566

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 3558	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝜓 → ∃𝑥∃𝑦𝜑))
5	1, 2, 4	mp2an 702	1 ⊢ (𝜓 → ∃𝑥∃𝑦𝜑)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559  ∃wex 1798   ∈ wcel 2141  Vcvv 3453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-clel 2836

This theorem is referenced by:  relop  5820  endisj  9032  dcomex  10401  axcnre  11119  hashle2pr  14487  wlk2f  29776  uhgr3cyclex  30330  qqhval2  34240  satfv1  35677  itg2addnclem3  38136  funop1  47841  cycldlenngric  48514  lgricngricex  48715