Theorem cbvexsv 45297

Description: A theorem pertaining to the substitution for an existentially quantified variable when the substituted variable does not occur in the quantified wff. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
cbvexsv	⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)

Distinct variable group:   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem cbvexsv

Step	Hyp	Ref	Expression
1		cbvrexsv 3359	. 2 ⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑦 ∈ V [𝑦 / 𝑥]𝜑)
2		rexv 3485	. 2 ⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)
3		rexv 3485	. 2 ⊢ (∃𝑦 ∈ V [𝑦 / 𝑥]𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
4	1, 2, 3	3bitr3i 304	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)

This proof depends on syntax axioms:   ↔ wb 209  ∃wex 1812  [wsb 2099  ∃wrex 3092  Vcvv 3458

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-13 2407  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ral 3083  df-rex 3093  df-v 3460

This theorem is used by:  onfrALTlem1  45298  onfrALTlem1VD  45639