Theorem mobidv 2479

Description: Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by Mario Carneiro, 7-Oct-2016.)

Hypothesis

Ref	Expression
mobidv.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
mobidv	⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem mobidv

Step	Hyp	Ref	Expression
1		nfv 1830	. 2 ⊢ Ⅎ𝑥𝜑
2		mobidv.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	1, 2	mobid 2477	1 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 195  ∃*wmo 2459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-ex 1696  df-nf 1701  df-eu 2462  df-mo 2463

This theorem is referenced by:  mobii  2481  mosubopt  4897  dffun6f  5818  funmo  5820  caovmo  6769  1stconst  7152  2ndconst  7153  brdom3  9231  brdom6disj  9235  imasaddfnlem  16011  imasvscafn  16020  hausflim  21595  hausflf  21611  cnextfun  21678  haustsms  21749  limcmo  23452  perfdvf  23473  phpreu  32563  funressnfv  39857