Theorem mobidv 2091

Description: Formula-building rule for "at most one" quantifier (deduction form). (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 1552	. 2 ⊢ Ⅎ𝑥𝜑
2		mobidv.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	1, 2	mobid 2090	1 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 105  ∃*wmo 2056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-17 1550  ax-ial 1558

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-eu 2058  df-mo 2059

This theorem is referenced by:  mobii  2092  mosubopt  4744  dffun6f  5289  funmo  5291  1stconst  6314  2ndconst  6315  exmidmotap  7380  imasaddfnlemg  13190  dvfgg  15204