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Mirrors > Home > ILE Home > Th. List > 4exbidv | GIF version |
Description: Formula-building rule for 4 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 1841 | . 2 ⊢ (𝜑 → (∃𝑧∃𝑤𝜓 ↔ ∃𝑧∃𝑤𝜒)) |
3 | 2 | 2exbidv 1841 | 1 ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧∃𝑤𝜓 ↔ ∃𝑥∃𝑦∃𝑧∃𝑤𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∃wex 1469 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1424 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-4 1488 ax-17 1507 ax-ial 1515 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: ceqsex8v 2734 copsex4g 4177 opbrop 4626 ovi3 5915 brecop 6527 th3q 6542 dfplpq2 7186 dfmpq2 7187 enq0sym 7264 enq0ref 7265 enq0tr 7266 enq0breq 7268 addnq0mo 7279 mulnq0mo 7280 addnnnq0 7281 mulnnnq0 7282 addsrmo 7575 mulsrmo 7576 addsrpr 7577 mulsrpr 7578 |
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