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| Mirrors > Home > MPE Home > Th. List > 4exbidv | Structured version Visualization version GIF version | ||
| Description: Formula-building rule for four existential quantifiers (deduction form). (Contributed by NM, 3-Aug-1995.) | 
| Ref | Expression | 
|---|---|
| 4exbidv.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | 
| Ref | Expression | 
|---|---|
| 4exbidv | ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧∃𝑤𝜓 ↔ ∃𝑥∃𝑦∃𝑧∃𝑤𝜒)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 4exbidv.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
| 2 | 1 | 2exbidv 1924 | . 2 ⊢ (𝜑 → (∃𝑧∃𝑤𝜓 ↔ ∃𝑧∃𝑤𝜒)) | 
| 3 | 2 | 2exbidv 1924 | 1 ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧∃𝑤𝜓 ↔ ∃𝑥∃𝑦∃𝑧∃𝑤𝜒)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∃wex 1779 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 | 
| This theorem depends on definitions: df-bi 207 df-ex 1780 | 
| This theorem is referenced by: ceqsex8v 3540 copsex4g 5500 opbrop 5783 ov3 7596 brecop 8850 addsrmo 11113 mulsrmo 11114 addsrpr 11115 mulsrpr 11116 dihopelvalcpre 41250 xihopellsmN 41256 dihopellsm 41257 | 
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