Theorem r19.28mv 3517

Description: Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.)

Assertion

Ref	Expression
r19.28mv	⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)))

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem r19.28mv

Step	Hyp	Ref	Expression
1		nfv 1528	. 2 ⊢ Ⅎ𝑥𝜑
2	1	r19.28m 3514	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∃wex 1492   ∈ wcel 2148  ∀wral 2455

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-cleq 2170  df-clel 2173  df-ral 2460

This theorem is referenced by:  iinrabm  3951  iindif2m  3956  iinin2m  3957  xpiindim  4766  fintm  5403  ixpiinm  6726  neipsm  13739