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Mirrors > Home > ILE Home > Th. List > 19.8a | GIF version |
Description: If a wff is true, then it is true for at least one instance. Special case of Theorem 19.8 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
19.8a | ⊢ (𝜑 → ∃𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . . 3 ⊢ (∃𝑥𝜑 → ∃𝑥𝜑) | |
2 | hbe1 1495 | . . . 4 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑) | |
3 | 2 | 19.23h 1498 | . . 3 ⊢ (∀𝑥(𝜑 → ∃𝑥𝜑) ↔ (∃𝑥𝜑 → ∃𝑥𝜑)) |
4 | 1, 3 | mpbir 146 | . 2 ⊢ ∀𝑥(𝜑 → ∃𝑥𝜑) |
5 | 4 | spi 1536 | 1 ⊢ (𝜑 → ∃𝑥𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1351 ∃wex 1492 |
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-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-4 1510 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: 19.8ad 1591 19.23bi 1592 exim 1599 19.43 1628 hbex 1636 19.2 1638 19.9t 1642 19.9h 1643 excomim 1663 19.38 1676 nexr 1692 sbequ1 1768 equs5e 1795 exdistrfor 1800 sbcof2 1810 mo2n 2054 euor2 2084 2moex 2112 2euex 2113 2moswapdc 2116 2exeu 2118 rspe 2526 rsp2e 2528 ceqex 2864 vn0m 3434 intab 3873 copsexg 4244 eusv2nf 4456 dmcosseq 4898 dminss 5043 imainss 5044 relssdmrn 5149 oprabid 5906 tfrlemibxssdm 6327 tfr1onlembxssdm 6343 tfrcllembxssdm 6356 snexxph 6948 nqprl 7549 nqpru 7550 ltsopr 7594 ltexprlemm 7598 recexprlemopl 7623 recexprlemopu 7625 suplocexprlemrl 7715 |
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