Theorem r19.23v 2425

Description: Theorem 19.23 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.)

Assertion

Ref	Expression
r19.23v	⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓))

Distinct variable group:   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem r19.23v

Step	Hyp	Ref	Expression
1		nfv 1421	. 2 ⊢ Ⅎ𝑥𝜓
2	1	r19.23 2424	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 98  ∀wral 2306  ∃wrex 2307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-ral 2311  df-rex 2312

This theorem is referenced by:  uniiunlem  3028  dfiin2g  3690  iunss  3698  ralxfr2d  4196  rexxfr2d  4197  ssrel2  4430  reliun  4458  funimaexglem  4982  funimass4  5224  ralrnmpt2  5615