Theorem spc2ev 2947

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 2941	. 2 ⊢ ((A ∈ V ∧ B ∈ V) → (ψ → ∃x∃yφ))
5	1, 2, 4	mp2an 653	1 ⊢ (ψ → ∃x∃yφ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861

This theorem is referenced by:  opkabssvvk  4208  dfpw12  4301  pw1equn  4331  pw1eqadj  4332  dfxp2  5113  brtxp  5783  endisj  6051  mucnc  6131  ce0nnul  6177  cenc  6181  ce2  6192