Theorem mobidv 2078

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 1539	. 2 ⊢ Ⅎ𝑥𝜑
2		mobidv.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	1, 2	mobid 2077	1 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 105  ∃*wmo 2043

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-eu 2045  df-mo 2046

This theorem is referenced by:  mobii  2079  mosubopt  4724  dffun6f  5267  funmo  5269  1stconst  6274  2ndconst  6275  exmidmotap  7321  imasaddfnlemg  12897  dvfgg  14842