Theorem spc2ev 2728

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 2722	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝜓 → ∃𝑥∃𝑦𝜑))
5	1, 2, 4	mp2an 418	1 ⊢ (𝜓 → ∃𝑥∃𝑦𝜑)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1296  ∃wex 1433   ∈ wcel 1445  Vcvv 2633

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-v 2635

This theorem is referenced by:  relop  4617  th3qlem2  6435  endisj  6620  axcnre  7513