Theorem spc2ev 2826

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 2820	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝜓 → ∃𝑥∃𝑦𝜑))
5	1, 2, 4	mp2an 424	1 ⊢ (𝜓 → ∃𝑥∃𝑦𝜑)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348  ∃wex 1485   ∈ wcel 2141  Vcvv 2730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732

This theorem is referenced by:  relop  4761  th3qlem2  6616  endisj  6802  axcnre  7843