Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > r19.43 | Structured version Visualization version GIF version |
Description: Restricted quantifier version of 19.43 1883. (Contributed by NM, 27-May-1998.) (Proof shortened by Andrew Salmon, 30-May-2011.) |
Ref | Expression |
---|---|
r19.43 | ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.35 3295 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (¬ 𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 ¬ 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)) | |
2 | df-or 845 | . . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓)) | |
3 | 2 | rexbii 3210 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ ∃𝑥 ∈ 𝐴 (¬ 𝜑 → 𝜓)) |
4 | df-or 845 | . . 3 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓) ↔ (¬ ∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)) | |
5 | ralnex 3199 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜑) | |
6 | 5 | imbi1i 353 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 ¬ 𝜑 → ∃𝑥 ∈ 𝐴 𝜓) ↔ (¬ ∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)) |
7 | 4, 6 | bitr4i 281 | . 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓) ↔ (∀𝑥 ∈ 𝐴 ¬ 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)) |
8 | 1, 3, 7 | 3bitr4i 306 | 1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∨ wo 844 ∀wral 3106 ∃wrex 3107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-ral 3111 df-rex 3112 |
This theorem is referenced by: r19.44v 3305 r19.45v 3306 r19.45zv 4406 r19.44zv 4407 iunun 4978 wemapsolem 8998 pythagtriplem2 16144 pythagtrip 16161 dcubic 25432 legtrid 26385 axcontlem4 26761 erdszelem11 32561 satfvsucsuc 32725 fmla1 32747 soseq 33209 seglelin 33690 diophun 39714 rexzrexnn0 39745 ldepslinc 44918 |
Copyright terms: Public domain | W3C validator |